STABILITY and EXISTENCE of SURFACES in SYMPLECTIC 4-MANIFOLDS with B+ = 1

STABILITY AND EXISTENCE OF SURFACES IN SYMPLECTIC 4-MANIFOLDS WITH b+ = 1

JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

Abstract. We establish various stability results for symplectic surfaces in symplectic 4−manifolds with b+ = 1. These results are then applied to prove the existence of representatives of Lagrangian ADE- conﬁgurations as well as to classify negative symplectic spheres in symplectic 4−manifolds with κ = −∞. This involves the explicit construction of spheres in rational manifolds via a new construction technique called the tilted transport.

MSC classes: 53D05, 53D12, 57R17, 57R95 Keywords: smooth spheres, symplectic spheres, Lagrangian submanifolds

Contents 1. Introduction 2 2. A Technical Existence Result 7 2.1. Generic Almost Complex Structures 8 2.2. Existence of Symplectic Submanifold 10 2.3. The Relative Symplectic Cone 14 3. Stability of symplectic curve configurations 16 3.1. Smoothly isotopic surfaces under deformation–Theorem 1.3 16 3.2. Existence of Diffeomorphic surfaces 18 4. Lagrangian ADE-configurations 20 4.1. Conifold transitions and stability of Lagrangian configurations 20 4.2. Existence 23 5. Spheres in Rational Manifolds 25 5.1. Homology classes of smooth −4 spheres 25 5.2. Tilted transport: constructing symplectic (-4)-spheres 33 5.3. Spheres with self-intersection −1,−2 and −3 37 5.4. Discussions 39 6. Spheres in irrational ruled manifolds 41 6.1. Smooth spheres 41 6.2. Symplectic spheres 44

JD was partially supported by the Simons Foundation #246043. TJL was partially supported by was partially supported by NSF Focused Research Grants DMS-0244663 and NSF grant DMS-1207037. WW was partially supported by NSF Focused Research Grants DMS-0244663 and AMS-Simons travel funds. 1 2 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

References 45

1. Introduction Given a symplectic manifold (M, ω), it is natural to ask whether a homology/cohomology class A is represented by an embedded symplectic (La- grangian) submanifold. Even with the various advanced techniques currently available and emerging nowadays, this remains a very difficult question. There are two particularly significant techniques in this direction: for the classes l[ω] when [ω] has integral period and l is a sufficiently high multiple, a general existence was obtained by Donaldson’s asymptotic holomorphic section theory; for a homology class A ∈ H2(M, Z) which is Gromov-Witten effective, the pseudo-holomorphic curve machinery often produces embedded symplectic representatives in this class. In dimension 4, Taubes’ symplectic Seiberg-Witten theory [48, 49, 50] is especially powerful to establish the GW effectiveness. In the current paper, we investigate cases in dimension 4 not covered by the techniques mentioned above, e.g. we consider GW non-effective (or more precisely, not necessarily GW effective) classes. In fact, we approach this problem by answering a natural extension that is closely related but rarely seen in the literature: If there exists V ⊂ M which is an ω-symplectic submanifold, can V be “propagated” to other symplectic forms ωe? Such “propagation” can be interpreted in different senses. For example, when ωe is isotopic to ω, this problem has no new content due to Moser’s theorem. The main case we consider is when ωe is deformation equivalent to ω, that is, when there is a smooth family of symplectic forms {ωt} such that ω0 = ω and ω1 = ωe. When such propagation holds, we say (M, V, ω) possesses the stability property. In this paper we establish several stability results for connected symplectic surfaces in symplectic manifolds (M, ω) with b+ = 1, which allow us to address also the existence problem in rather general settings. To explain further our results, we first introduce some notions.

Definition 1.1. Consider a graph G, where each vertex vi is labelled by an element Ai ∈ H2(M, Z), and we denote aij := Ai · Aj ≥ 0. Two vertices are connected by edges labeled by positive integers which sum up to aij when aij 6= 0 . Assume that |G| < ∞. We will refer to G as a homological configuration. Let ω be a symplectic structure on M. (1) G is called simple if all labels on the edges are 1. (2) ω is called G-positive if ω(Ai) > 0 for all i ≤ |G|. S|G| (3) A curve configuration V = i=1 Vi is a realization of the homological configuration G with respect to ω, if it consists of the following: STABILITY AND EXISTENCE OF SURFACES 3

(a) a one-one correspondence from the vertices {vi} of G to an embedded ω-symplectic curve Vi ⊂ M, [Vi] = Ai for each i ≤ |G| where Ai is the homology class labeled on vi; (b) a one-one correspondence from Vi ∩ Vj to the edges connecting vi and vj, and the intersection multiplicity equals the marking on the corresponding edges; (c) Vi ∩ Vj ∩ Vk = ∅ for all distinct i, j, k and (d) there exists an almost complex structure J compatible with ω making each Vi J-holomorphic simultaneously. Notice that the curve configuration need not be connected. Moreover, the last condition ensures that all intersections of components of V are isolated and positive. We consider stability for such configurations. Definition 1.2. A curve configuration V realizing G with respect to ω is called ω-stable if for any G-positive symplectic form ω˜ deformation equivalent to ω, there is a curve configuration V˜ realizing G with respect to ω˜. In some cases the relation between V˜ and V can be made more precise. The following is the main stability result. Theorem 1.3. Let (M, ω) be a symplectic manifold with b+ = 1 and G a homological configuration represented by a curve configuration V . Then V is ω-stable. Moreover, V˜ can be chosen to be smoothly isotopic to V . At the core of Theorem 1.3 are the existence and abundance of positive self-intersection symplectic surfaces along which inflation is carried out. The major source of such surfaces is Taubes’ symplectic Seiberg-Witten theory. Moreover, the methods employed to prove Theorem 1.3 are rather robust and allow extensions in a number of directions. We describe details in Sections 2 and 3. In the rest of the paper we consider two applications of the stability result Theorem 1.3. The first one is to show the following: Corollary 1.4. In rational or ruled manifolds, any homological Lagrangian n ADE-configuration {li}i=1 admits a Lagrangian ADE-configuration representative. In the case of An-configurations, one may require the configuration lie in M\D, where D is a symplectic divisor disjoint from a set of n+1 embedded symplectic representatives of the exceptional classes {Ei}i=1 The definition of a homological Lagrangian configuration is given in Sec- tion 4. For more general symplectic manifolds with b+ = 1 we have: Corollary 1.5. Given a non-minimal symplectic 4-manifold (M, ω) with + n+1 b = 1 and a set of exceptional classes {Ei}i=1 where ω(Ei) are all equal. n Then there is a Lagrangian An-configuration of class {Ei − Ei+1}i=1. It is very tempting to assert the above corollary also holds for general symplectic 4-manifolds. But there is a (possibly technical) catch: in general 4 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU we do not know whether two ball embeddings in a general symplectic 4- manifold are connected. That means it is possible that two symplectic blow- up forms are not even symplectomorphic. Therefore one needs to be more precise when performing symplectic blow-ups on these manifolds. Recall from Biran’s stability of ball-packing in dimension 4 ([4]) that for any symplectic 4−manifold (M, ω), there exists a number N0(M, ω) so that one may pack N balls with volume less than volω(M)/N0, as long as [ω] ∈ H2(M, Q). The packing is constructed away from a isotropic skeleton defined in [5]. We show that An-type configurations still exist when these packed symplectic balls are blown-up.

Corollary 1.6. Given a symplectic 4-manifold (M, ω) with [ω] ∈ H2(M, Q) and a symplectic packing of n + 1 ≤ N0 symplectic balls with equal volume ≤ volω(M)/N0(M, ω). Then there is a Lagrangian An-conﬁguration in (M#(n + 1)CP2, ω0), where ω0 is obtained by blowing up the embedded symplectic balls when the packing is supported away from Biran’s isotropic skeleton.

Remark 1.7. The more interesting part of this series of corollaries lies in the case when the packing of M is very close to a full packing. In such scenarios, the geometry of the packing is usually difficult to understand in an explicit way. In particular, it would be very difficult to construct these La- grangian spheres by hand. In contrast, our theorem does not only guarantee the existence of the Lagrangian spheres (which already appeared in [34]), we also have control over their geometric intersection patterns, which is usually difficult for Lagrangian or symplectic non-effective objects. From our proof, one may also conclude the existence of symplectic ADE-plumbings when the involved classes have positive symplectic areas (in fact, this is much easier because we do not need to involve conifold transitions and can easily be extended to many other types of plumbings). We leave the details for interested readers.

As another application, we consider the classification of negative self- intersection spherical classes in symplectic rational or ruled surfaces. This is of interest for many different reasons: on the one hand, solely the problem of existence of symplectic rational curves is already an intriguing question when the corresponding class is not GT-basic, which means its Gromov- Taubes dimension is less than zero. However, such curves are exactly the most interesting objects in many areas of research. For example, they span the Mori cone in birational geometry, which has been extended into the symplectic category. Moreover, such rational curves and the configurations they form are crucial for various constructions in symplectic geometry ( for a very incomplete list, see [18], [46, 47], [45], [1], [23]). As we will describe, we have found (-4)- symplectic spheres in CP 2#10CP 2 along which a rational blow-down incurs STABILITY AND EXISTENCE OF SURFACES 5 exotic examples of E(1)2,k. This will be exploited further in our upcoming work [DLW]. In our approach, we also solved the problem of classifying homology classes of smooth embedded (−4)-spheres in rational manifolds, which, to the best of authors’ knowledge, is also new to the literature. To state our result, denote the geometric automorphism group by + (1.1) D(M) = {σ ∈ Aut(H2(M, Z)) : σ = f∗ for some f ∈ Diff (M)}.

Two classes A, B ∈ H2(M, Z) are called D(M)-equivalent if there is a σ ∈ D(M) such that σ(A) = B. The following is our main result in Section 5: Theorem 1.8 (Classification of −4-spheres). Let (M, ω) be a rational sym- 2 2 plectic surface, i.e. M = CP #kCP and {H,E1, ..., Ek} the standard basis of H2(M, Z). Consider any class A ∈ H2(M, Z) with A · A = −4. • (Smooth case) A is represented by a smooth sphere if and only if A is D(M)-equivalent to one class in the following list (1) −H + 2E1 − E2 (2) H − E1 − .. − E5 P9 (3) −a(−3H + i=1 Ei) − 2E10 for some a ∈ N and a ≥ 2 (4) 2E1 (5) 2(H − E1 − E2) • (Symplectic case) A is represented by an ω−symplectic sphere if and only if A is represented by a smooth sphere, [ω]·A > 0 and Kω ·A = 2 (Kω is the symplectic canonical class associated to ω). Moreover, up to D(M)-equivalence, the class A is one of the following: (i) If A is characteristic, then k = 5 and A is equivalent to type (2) above. (ii) If A is not characteristic, then it is equivalent to either type (1) or type (3) above. Notice that for large enough k, some of the classes listed above are in fact pairwise D(M)-equivalent. For completeness we will also give an overview of symplectic spheres of square −1, −2, −3 in rational manifolds. For those of squares −1, −2 the results are essentially contained in the earlier works [32] , [29] and [34]. We also provide an explicit algorithm in Remark 5.9 to implement our results. Our result should be considered preliminary as it leads to more interesting questions in two rather different directions. On the one hand, recall the bounded negativity conjecture asserts that any algebraic surface in charac- 2 teristic zero has C ≥ nX for any prime divisor C ⊂ X and some fixed nX ∈ Z. (for accounts on this conjecture in complex geometry, see for example [20], [3], see also [2] for variations on this conjecture). Note that even for rational manifolds, only certain small ones are known to satisfy this conjecture. 6 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

The conjecture makes perfect sense in the symplectic category, that is, whether squares of symplectic curves in a given symplectic manifold are bounded from below. For example, Lemma 6.1 partially reproduces the boundedness result of Prop. 2.1 in [3]. The computation relying on symplectic genus in 5.1.2 shows some preliminary dichotomy patterns for a negative curve: it is either not reduced (Section 5) and can be understood in small blow-ups, or its class is reduced but only appears for a relatively large number of blow-ups. For example, our result says (−4)-spheres can either be equivalent to a curve in two blow-ups, or its class may only appear when the blow-up number hits 10. It seems reasonable to speculate that this continues to hold at least for (−n)-spheres-there might be more complicated classes that are not blow-ups of classes in our list, but they only show up when the blow-up numbers are large enough, hence for a fixed symplectic rational manifold, there are only finitely many such hierarchies. If this could be verified, one could possibly approach the case of higher genus with similar methods. This will be investigated in future work. On the other hand, the existence part of Theorem 1.8 requires a new technique, which we call the tilted transport. This is very similar to the usual parallel transport construction of Lagrangian submanifolds, but due to the “softer” nature of symplectic objects, this construction is also much more flexible and even can be formulated quite combinatorially. Moreover, this simple technique could lead to the construction of a wealth of symplectic submanifolds that are not GW-effective out of a Lefschetz fibration, thus should be of independent interest. We describe this construction in Section 5.2 and apply it to construct (−4)-spheres in our classification. Similar to the case of rational manifolds, we obtain for irrational ruled manifolds the following more complete classification. A corresponding char- acterization in the smooth category appears as Lemma 6.1. Theorem 1.9. Suppose (M, ω) is an irrational ruled 4−manifold. Let A ∈ H2(M, Z) with ω(A) > 0. Then A is represented by a connected ω−symplectic sphere if and only if A is represented by a connected smooth sphere and gω(A) = 0. Moreover, suppose A is represented by a connected ω-symplectic sphere. Then (1) A · A ≥ 1 − b−(M). (2) A is characteristic only if A·A = 1−b−(M), and A is D(M)−equivalent to E1 − E2 − · · · − E1−b−(M). (3) If A is not characteristic, then A is D(M)−equivalent to F − E1 − · · · − El for l = −A · A. Moreover, when A is not characteristic, then it is the blow-up of an exceptional sphere. As stated above, one motivation for investigating stability is to find a general existence criterion for connected embedded symplectic surfaces in a given homology class (as discussed in the survey [30], see also [35]) although STABILITY AND EXISTENCE OF SURFACES 7 we have generalized the context to symplectic configurations. The results above lead us to offer the following speculation.

Speculation 1.10. Let (M, ω) be a symplectic 4-manifold. Let A ∈ H2(M, Z) be a homology class. Then A is represented by a connected ω-symplectic surface if and only if (1) [ω] · A > 0, (2) gω(A) ≥ 0 and (3) A is represented by a smooth connected surface of genus gω(A). Theorem 1.8 thus verifies this speculation for spheres in rational manifolds with A2 ≥ −4 and Theorem 1.9 for all spheres in irrational ruled surfaces. This is in a sense by “brutal force”: we give a complete classifications of the smoothly representable and symplectically representable classes and compare them. It would be interesting to have a construction independent of these classification results.

Outline of the paper: In Section 2 and 3 we establish the stability result 1.3. Section 2 contains some technical tools useful for finding symplectic submanifolds and inflations adapted from [11] to the configuration case. In Section 4 we consider the stability and existence for Lagrangian configurations. Section 5 classifies (−4)-spheres in both smooth and symplectic categories, with a subsection specifically devoted to tilted transports. Sec- tion 6 completes the discussion for symplectic manifolds with κ(M) = −∞ by considering irrational ruled manifolds.

2 2 Notation: Let M = CP #kCP . We use the standard basis for H2(M, Z) Pk given by {H,E1,...,Ek}. Denote by Kst = −3H + i=1 Ei the standard 2 canonical class. Similarly we use the standard basis {A, B} for H2(S × S2, Z). For non-minimal irrational ruled surfaces M, we use {S, F, E1,...,En} as the basis of H2(M, Z), where S denotes the class of the base and F the class of the ﬁber.

Acknowledgements: The third author is grateful to Ronald Fintushel for introducing him to the problem of bounded negativity, and Kaoru Ono for explaining patiently many details regarding conifold transitions. He would also like to warmly thank Selman Akbulut for his interest in this work and oﬀering an opportunity to present it in the Topology seminar at MSU.

2. A Technical Existence Result In this section we wish to extend Theorem 2.13, [11], to the more complicated curve configurations of Def 1.1. The key to the proof of Theorem 2.13 is Lemma 2.14 therein, which provides for the existence of a curve in a given class A ∈ H2(M; Z) under certain restrictions on A. At the core 8 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU of the proof of this lemma are results on the existence of a suitably generic almost complex structure among those making a fixed submanifold V pseudoholomorphic such that classes A with negative Gromov dimension are not represented by pseudoholomorphic curves. k Consider a curve configuration V = ∪i=1Vi. Let JVi denote the set of almost complex structures compatible with ω and making Vi pseudoholomorphic and let JV = ∩iJVi . Notice that JV 6= ∅ by Def 1.1. 2.1. Generic Almost Complex Structures. We begin by defining a universal space which we shall use throughout this section: Fix a closed compact Riemann surface Σ. The universal model U is defined as follows: this space will consist of Diff(Σ) orbits of a 4-tuple (i, u, J, Ω) with (1) u :Σ → M an embedding off a finite set of points from a Riemann k,p surface Σ such that u∗[Σ] = A ∈ H2(M, Z) and u ∈ W (Σ,M) with kp > 2, 1 (2) Ω ⊂ M a set of k(A) = 2 (A · A − Kω · A) distinct points (with Ω = ∅ if k(A) ≤ 0) such that Ω ⊂ u(Σ), (3) i a complex structure on Σ and J ∈ JV . Note that every map u is locally injective, and one has a fibration π : U → JV . Moreover, in order for the set U to be of any interest, it is natural to implicitly assume that A · [Vi] ≥ 0 for all i ∈ I unless A = [Vi] and [Vi] · [Vi] < 0. The goal of this section is to show that for a sufficiently generic choice of almost complex structure in JV the fiber in U either has the expected dimension or dim ker(π) = 0. We must distinguish two cases: If A 6= [Vi], then for any point in U, u(Σ) will contain a point not in V . If A = [Vi] for some i, then in U we will distinguish the embedding of Vi (and possibly Vj if [Vi] = [Vj]) from the other points in U. It should be noted that we prescribe V , hence the manifolds comprising V may be very poorly behaved with respect to Gromov-Witten moduli. In particular, k([Vi]) < 0 is possible. Consider first a class A 6= [Vi]. Note that this includes, for example, the class A = [Vi] + [Vj]. For such a class, any element u ∈ U will have a point x0 ∈ Σ such that u(x0) 6∈ Vi. Therefore, the proof of Lemma A.1, [11], applies as written there, albeit with a different set of underlying almost complex structures.

Lemma 2.1. Let A ∈ H2(M, Z), A 6= [Vi] for any i and k(A) ≥ 0. Let Ω denote a set of k(A) distinct points in M. Denote the set of pairs k(A) A (J, Ω) ∈ JV × M by I. Let JV be the subset of pairs (J, Ω) which are nondegenerate for the class A in the sense of Taubes [50, Def. 2.1]. A Then JV is a set of second category in I. We remind the readers that the nondegeneracy involved in the above lemma pertains only to embedded curves. Consider now the case A = [Vi] for some i (for simplicity assume i = 1). In this case, as noted above, U STABILITY AND EXISTENCE OF SURFACES 9 consists of two types of points: Those which are embeddings of components of V and those which contain a point not in V . We concentrate ﬁrst on the points corresponding to embeddings of components of V . Let ji be an almost complex structure on Vi and denote j = (j1, .., jk). Deﬁne j JV = {J ∈ JV |J|Vi = ji} j and call any J-holomorphic embedding for J ∈ JV a j-holomorphic embedding. Notice that j JV = ∪jJV j and by assumption JV 6= ∅. Hence for some j , JV 6= ∅. Consider the behavior of the linearization of ∂I,J at a point u ∈ U such that u : (Σ,I) → (M,J) is a j-holomorphic embedding of V1.

Lemma 2.2. Let A = [V1] and ﬁx j. Fix a j-holomorphic embedding u : (Σ,I) → (M,J) of V1 (or of Vj if [V1] = [Vj]). j j (1) If k(A) ≥ 0, then there exists a set GV of second category in JV such j that for any J ∈ GV the linearization of ∂i,J at the embedding u is surjective. j j (2) If k(A) < 0, then there exists a set GV of second category in JV such j that for any J ∈ GV the linearization of ∂i,J at the embedding u is injective. The proof of this lemma follows exactly as the proof of Lemma A.2, [11], as the necessary perturbations occur in a neighborhood of a point on V1 which is not contained in any other Vj. Our conditions ensure that such a point exists. The key point of Lemma 2.2 is that, in spite of the non-genericity of almost complex structure in JV , we may at least require that non-generic curves do not have nontrivial deformations. This is recapped in the following:

Lemma 2.3. Assume A = [Vi] for some i. Let Ω denote a set of k(A) distinct points in M. k(A) (1) k(A) ≥ 0: Denote the set of pairs (J, Ω) ∈ JV × M by I. Let J[V ] be the subset of pairs (J, Ω) which are nondegenerate for the class A in the sense of Taubes [50]. Then J[V ] is dense in I. (2) k(A) < 0: There exists a set of second category J[V ] ⊂ JV such that there exist no pseudoholomorphic deformations of V and there are no other pseudoholomorphic maps in class A except possibly components of V . The general tactic for a proof of this lemma is as follows: Recall that ³ ´ k(A) j k(A) JV × M = tj JV × M . 10 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

j k(A) Now find a dense subset of JV × M (when non-empty). Lemma 2.2 j j provides for a suitable subset GV ⊂ JV of second category for each embedding of a component of V ensuring that the differential operators at such an embedding have the appropriate behavior. Taking the intersection of all j such sets produces a set GV which is still of second category in JV . Now consider only almost complex structures J ∈ GV to understand the behavior of U at points which have a point in the image off of V . The methods of the proof of Lemma A.1, [11] apply in this setting. 2.2. Existence of Symplectic Submanifold. In this section we state and justify a result analogous to Lemma 2.14, [11]. Let V be a realization of some homological configuration G. Note that Lemma 2.14, [11], can be used to provide a ω-symplectic submanifold intersecting some Vi as needed, however it is not immediatley clear why this curve must intersect the other Vi also locally positively and transversally. In particular, the restriction of almost complex structures from JVi to JV must be justified. This has been prepared in the previous section and at all points in the proof of Lemma 2.14, [11], these results should be inserted. We begin with the following observation. Lemma 2.4. Let (M, ω) be a symplectic manifold with b+(M) = 1, W a connected embedded symplectic submanifold and A ∈ H2(M, Z). Assume that (A − Kω) · [W ] > 0, A · A ≥ 0 and A · [ω] ≥ 0. Then (1) A · [W ] ≥ 0 and (2) if A · [W ] = 0, then either [W ] · [W ] = 0 and A = λ[W ] up to torsion or W is an exceptional sphere. Proof. If [W ] · [W ] ≥ 0, then A · [W ] ≥ 0 by the light cone lemma (Lemma 3.1, [32]). Now consider [W ] · [W ] < 0. Let (A − Kω) · [W ] > 0 and assume further that A · [W ] < 0. Then

Kω · [W ] < A · [W ] < 0. As W is a connected embedded symplectic submanifold, it satisﬁes the adjunction equality which implies

[W ] · [W ] + 2 − 2g = −Kω · [W ] > 0 and thus [W ] · [W ] > 2g − 2. Therefore [W ] · [W ] ≥ 0 unless g = 0 and [W ] · [W ] = −1. Thus [W ] is an exceptional sphere. Then −1 = Kω · [W ] < A · [W ] < 0, contradicting A · [W ] < 0. This proves the non-negativity statement of the lemma. Assume that A · [W ] = 0. Then by the light cone lemma [W ] · [W ] ≤ 0. If [W ] · [W ] = 0, then, again by the light cone lemma, A = λ[W ] up to torsion. Otherwise W is an exceptional sphere. ¤ The following is a version of this statement for exceptional spheres. STABILITY AND EXISTENCE OF SURFACES 11

Lemma 2.5. Let (M, ω) be a symplectic manifold with b+(M) = 1, W an embedded symplectic submanifold and A ∈ Eω an exceptional sphere. Let (A − Kω) · [W ] > 0. Then (1) A · [W ] ≥ 0 unless A = [W ] and (2) if A · [W ] = 0, then there exists an exceptional sphere in the class of A which is disjoint from W . Proof. Notice that A is GT-basic as it is an exceptional sphere. Assume that [W ] · [W ] ≥ 0. Then for any almost complex structure making W pseudoholomorphic, we can ﬁnd a connected pseudoholomorphic representative for A. This curve may have many components connected by nodes, some multiply covered, but each image curve must intersect W locally positively. In particular, this representative of A can have components covering W , however these also contribute only positively to A · [W ]. Therefore A · [W ] ≥ 0. Let [W ] · [W ] < 0. As in Lemma 2.4 , the assumption (A − Kω) · [W ] > 0 and A · [W ] < 0 implies that W is an exceptional sphere. Lemma 3.5, [32], ensures that A · [W ] ≥ 0 unless A = [W ]. Let A·[W ] = 0. If [W ]·[W ] ≥ 0, then by the above argumentP a connected J-holomorphic representative of A can be found such that A = Ai+m[W ]. Since each component Ai intersects W non-negatively, by pairing with [W ], one sees that Ai are indeed disjoint from W . The connectedness assumption thus implies m = 0. A standard genericity argument shows by perturbing J away from W we may assume there is only one component among Ai which is non-empty, giving the desired exceptional sphere. If W is an exceptional sphere, then Lemma 3.5, [32], provides for the existence of a representative of A disjoint from W . ¤

The converse of these results, i.e. that A·[W ] ≥ 0 implies (A−Kω)·[W ] > 2 0, need not be true. Let M = S × Σ3,Σ3 a genus 3 surface. Consider the standard basis of H2(M, Z) and any symplectic form with Kω = 4F − 2S. Let [W ] = S −F and A = S +F . Then A·[W ] = 0 but (A−Kω)·[W ] = −6. Notice that [W ] is representable by a symplectic submanifold of genus 3 for some symplectic form with this canonical class. The following result is an extension of Lemma 2.14, [11] from a submanifold to a curve conﬁguration. The proof is largely identical hence we only give an outline with appropriate details relevant to multiple components. Lemma 2.6. Fix a symplectic form ω on M with b+(M) = 1 such that V is a curve conﬁguration. For any A ∈ H2(M; Z) with

A · E > 0 for all E ∈ Eω, A · A > 0,A · [ω] > 0,

(A − Kω) · [ω] > 0, (A − Kω) · (A − Kω) > 0,

(A − Kω) · [Vi] > 0 for all i ∈ I, 12 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU there exists a connected embedded ω-symplectic submanifold C in the class A, intersecting V ω-orthogonally and positively. Proof. The assumptions

A · E > 0 for all E ∈ Eω, A · A > 0,A · [ω] > 0,

(A − Kω) · [ω] > 0, (A − Kω) · (A − Kω) > 0, together with b+ = 1 ensure that for generic almost complex structures A admits a connected embedded pseudoholomorphic representative (see [32]). By Lemma 2.4, the assumption (A−Kω)·[Vi] > 0 together with A·A > 0 and A·[ω] > 0 ensures that A·[Vi] > 0 unless possibly if Vi is an exceptional sphere. In the latter case, A · [Vi] > 0 by assumption. Therefore the results of the previous section on the genericity of almost complex structures can be applied to A and its components. Standard arguments (see for example the proof of Lemma 2.14, [11]) lead to the following decomposition for the class A: X X X X A = mi[Vi] + mi[Vi] + biBi + aiAi

k([Vi])≥0 k([Vi])<0 i i where

(1) Bi ∈ Eω, (2) all intersections of distinct classes are non-negative, (3) Ai · Ai ≥ 0 and k(Ai) ≥ 0 and (4) all sums are ﬁnite.

Assume that the second summand (over k([Vi]) < 0) is empty. Then consider any Vi with A · [Vi] > 0 and ﬁx this as Vi = V0 ( in particular, if A = m[Vi], choose this particular i). Move all of the other [Vi]-terms into either the Bi or the Ai summand as appropriate (this can be done as k([Vi]) ≥ 0). Now repeat the argument in Case 1 of the proof of Lemma 2.14, [11]. As in that proof, if A 6= m[V0] or A = m[V0] and k([V0]) > 0, then the proof is complete. The remaining case is A = m[V0] and k([V0]) = 0, 2 2 which implies that either m = 1 or [V0] = 0. Notice that if [V0] = 0, then A · [V0] = 0, contradicting our assumption that A · A > 0. If m = 1, then A = [V0] and thus (A − Kω) · [V0] = 2k([V0]) = 0, contradicting our assumptions. This provides for an embedded J-holomorphic curve C˜ representing A with a single non-multiply covered component intersecting V0 positively where J is now chosen appropriately from JV . This however also implies that C˜ intersects all Vi locally positively. When A 6= m[V0], then also, by our choice of V0, A 6= n[Vi] for all i, and hence we can ensure that C˜ is distinct from any Vi. When k([V0]) > 0, then by an appropriate choice of ˜ points Ωk(A) we can again ensure that C is distinct from any Vi. Each of the members of V is distinct, hence all intersection points are isolated. Apply Lemma 3.2 and Prop. 3.3, [33], to perturb only C˜ to a STABILITY AND EXISTENCE OF SURFACES 13

0 pseudoholomorphic curve C , while leaving each Vi unchanged, such that C0 intersects the pseudoholomorphic curve family locally positively and transversally. Now perturb C0 further to a J-holomorphic curve C which is ω-orthogonal to V . This involves a local perturbation around the intersection points and can be done in such a way as to ensure that distinct intersections stay distinct, see [19]. Assume now that the second summand is notP empty. We rewrite the class A as follows. First, move all of the terms in m [V ] into either the k([Vi])≥0 i i PBi or the Ai summand, as done in the ﬁrst case. Secondly, distinguish in m [V ] those classes corresponding to components of the curve in a k([Vi])<0 i i class mi[Vi] which are not multiple covers of Vi and those which are multiple covers. As noted in [11], the former all correspond to curves which underlie the genericity results discussed previously. A generic choice of almost complex structure in JV thus either removes such curves (if k(mi[Vi]) < 0) or they can be included in the Ai or Bi sum (if k(mi[Vi]) ≥ 0). Denote the remaining terms X Vmult = mi[Vi].

k([Vi])<0, V mult. cover i P P Consider now A˜ = A − Vmult = aiAi + biBi. The arguments in [4] or [11] continue to hold albeit with mZ resp. mV replaced by Vmult. As in P 0 [4], we obtain the estimate k(Ai) ≤ k (A˜). Moreover, 2k(A) − 2k0(A˜) = (A − K ) · V + non-negative terms | {zω mult} >0 by assumption P Hence k(A) > k(Ai) and thus either A = Vmult or Vmult = 0. In the latter case the result follows from arguments as above. If A = Vmult, then note that X 2k(A) = A · A − K · A = mi(A − K) · [Vi] > 0.

k([Vi])<0, Vi mult. cover Thus choose a point not on V . Then by Lemma 2.1 and 2.3 together with the assumptions on A there exist (J, Ω) such that A is represented by an embedded curve meeting V locally positively, and, as before, this curve can be made ω-orthogonal to V by the results in [33] and [19]. ¤ Corollary 2.7. For any ω, A and V as in Lemma 2.6, there is a family of symplectic forms {ωt}0≤t≤1 such that ω0 = ω, [ωt] = [ω] + tP D([A]) and V is a curve conﬁguration with respect to ωt. Notice that the conditions A · E > 0 for all E ∈ Eω, (2.1) A · A > 0,A · [ω] > 0

(A − Kω) · [ω] > 0, (A − Kω) · (A − Kω) > 0 14 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

from Lemma 2.6 on A ensure that A has a J-holomorphic representative for any J. In fact, they ensure that A is a GT-basic class, see [32]. Moreover, when b+ = 1, this class is representable by a connected curve. In particular, the proof above makes explicit use of only A · [Vi] ≥ 0 and (A − Kω) · [Vi] > 0; any GT-basic classes satisfying these conditions will verify the lemma. Speciﬁcally, if A · A = −1, then it is necessary to assume that no component of V has [Vi] = A or a conclusion as in Lemma 2.6 must be false. However, this is implied by the assumptions (A − Kω) · [Vi] > 0 and A 6= [Vi] for all i ∈ I, see Lemma 2.5. Therefore the proof of Lemma 2.6 is immediately applicable, albeit with the slight change in Case 1 that A = Bi is now allowed. We state this as a lemma: Lemma 2.8. Fix a symplectic form ω on M such that V is a curve con- ﬁguration. Assume that A ∈ Eω and A 6= [Vi] for any i ∈ I. Furthermore, assume that

(A − Kω) · [Vi] > 0 for all i ∈ I. Then there exists a connected embedded ω-symplectic submanifold C in the class A intersecting V ω-orthogonally and positively. 2.3. The Relative Symplectic Cone. Let Ω(M) denote the space of orientation-compatible symplectic forms on M. The symplectic cone CM is the image of the cohomology class map Ω(M) → H2(M, R) ω 7→ [ω].

V For a smooth connected surface V , the relative symplectic cone CM ⊂ CM is the set of classes of symplectic forms making V a symplectic submanifold. V Since V is ω−symplectic, by Theorem 2.13 in [11], CM contains the cone A 0 0 0 CM,Kω = {α = [ω ]| ω symplectic with Kω = Kω, α · A > 0}, where A = [V ]. Now let V = ∪i∈I Vi be a collection of connected embedded curves. One V may similarly deﬁne CM . Let K be a symplectic canonical class for M and deﬁne

V DK = {[ω] ∈ CM | [ω] · [Vi] > 0 for all i ∈ I,Kω = K} .

This is the set of classes in the Kω-symplectic cone which pair positively with each component in the curve conﬁguration. By deﬁnition,

V [Vi] DK = ∩i∈I CM,K . Note that this does not imply the existence of a symplectic form ω making V a curve conﬁguration. STABILITY AND EXISTENCE OF SURFACES 15

+ Theorem 2.9. Let M be a 4-manifold with b (M) = 1 and {Vi}i∈I a family of submanifolds of M such that there exists a symplectic form ω on M making V = ∪Vi into a curve conﬁguration. Then

V V DKω ⊂ CM . In particular, for every α ∈ DV there exists a symplectic form τ in the Kω class α making V into a curve conﬁguration. Moreover, τ is deformation equivalent to ω through forms making V a curve conﬁguration.

Proof. Fix a symplectic form ω making V into a curve configuration. We 2 may assume that [ω] ∈ H (M, Z): since making Vi into a symplectic curve is an open condition and we have only finitely many components of V , we consider the intersection of these open sets. In this intersection there must be a symplectic form β with [β] ∈ H2(M, Q) making V into a curve configuration. Now rescale to get ω. Let e ∈ DV ∩ H (M, Z). Then the class A = le − PD[ω] satisfies Kω 2 the assumptions of Lemma 2.6 for sufficiently large l. Thus there exists an ω-symplectic submanifold C intersecting V locally positively and ω- orthogonally, with class [C] = A. 2 1 Let N be an S -bundle over a surface of genus gω(A) = 2 (A·A+Kω ·A)+1 and S be a section with S · S = −A · A. Now apply the pair-wise sum of Thm. 1.4, [19], to (M,C) and (N,S). The thus generated manifold X is diffeomorphic to M. This symplectic sum produces a family of deformation equivalent symplectic forms ωt on M in the class [ω]+tA for t ≥ 0 such that V is a curve configuration with respect to ωt. Thus [ω1] = le and the class e is representable by a symplectic form making V into a curve configuration. Now repeat the argument as in the proof of Theorem 2.13, [11] for general e ∈ DV . Kω ¤

Remark 2.10. The results of this section are similar to Theorem 1.2.7 (the second part), 1.2.12 and Corollary 1.2.13, [39] and it seems instructive to compare the two situations for the reader’s convenience. The key difference is in the conditions we imposed in Lemma 2.6, compared to the second part of Theorem 1.2.7 in [39] (we do not have a clear idea about the relations between the conditions therein and ours). Our set of assumptions, directly adapted from [11], offer several aspects of convenience. On the one hand, they replace rational/ruled assumptions in Proposition 3.2.3 and 5.1.6 in [39] so that we have results for manifolds with b+ = 1 with our assumptions. On the other hand, since this set of conditions is automatically satisfied when A is a positive class pairing with all components in the configurations positively and raised to a high multiple, they are particularly suitable for performing inflation and allows for slightly more flexibility. Hence our singular set places no restrictions on the intersections such as being transverse between components. 16 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

Another difference between the two results stem from the almost complex structures to be considered. In this paper we assume that there exists an almost complex structure making each Vi pseudoholomorphic at the same time. This statement concerns only the submanifolds themselves and the almost complex structure outside can be generically chosen. In [39] the authors consider adapted almost complex structures which places conditions on a fibered neighborhood of the configuration S. This is crucial for the geometric constructions therein, which was used to simplify the “B” part of the curve in the decomposition (3.1.2) in [39]. What we did not deal with in the current paper is the family inflation (Thm 1.2.12, [39]), which probably requires similar techniques as in [39]. Furthermore, the statement of Thm 2.9 is the same as Prop. 1.2.15(i), [39]. However, due to the differences in the sets under consideration, as described above, Thm 2.9 is in a more general setting, allowing more general manifolds and configurations. The proofs also differ slightly: [39] use inflation along nodal curves, we only need to consider embedded curves. Both results are an extension of Thm. 2.13, [11], which is for a single symplectic surface.

3. Stability of symplectic curve configurations Suppose that V is a symplectic surface in a symplectic 4-manifold (M, ω). Then we can consider the stability of this surface under (not necessarily continuous) variations of the symplectic structure. Our main result concerns deformations of the symplectic structure. This is the setting of Theorem 1.3 and is discussed in 3.1. Note that V and V˜ are not just homologous, but isotopic. In nice cases, we also address the stability for arbitrary symplectic struc- tureω ˜. The issue which arises in this context is that it is in general not understood how to go from one deformation class of symplectic structures to another. Two cases in which this is explicitly understood are the following. (1) The special family of manifolds with κ = −∞: For such manifolds, V and V˜ will be diﬀeomorphic, we treat this case in 3.2. (2) The special family of GT basic classes: Well known results imply that any surface V arising in this context is stable.

3.1. Smoothly isotopic surfaces under deformation–Theorem 1.3. With the preparatory work of the previous section, we are now ready to prove Theorem 1.3, which we recall here.

Theorem 3.1. Let (M, ω) be a symplectic manifold with b+(M) = 1 and G a homological conﬁguration represented by a curve conﬁguration V . Then V is ω-stable. Moreover, V˜ can be chosen such that each component of V˜ is smoothly isotopic to the corresponding (given by G) component in V . STABILITY AND EXISTENCE OF SURFACES 17

Proof. The assumption on V implies by Theorem 2.9 that DV ⊂ CV . Since Kω M ω˜ is deformation equivalent to ω and pairs positively with A, it follows that [˜ω] ∈ DV ⊂ CV , ie. there is a V −symplectic form τ cohomologous toω ˜. Kω M Notice that by Theorem 2.9 the V −relative symplectic forms are deformation equivalent to ω. Thus τ can be assumed to be deformation equivalent to ω. In [38] it is shown, that when b+ = 1, any deformation equivalent cohomologous symplectic forms are isotopic. It follows that τ andω ˜ are isotopic. Applying Moser’s Lemma, we obtain aω ˜−symplectic curve conﬁguration V˜ smoothly isotopic to V . ¤ Remark 3.2. (1) Note that it is not necessary to postulate that the deformation is through symplectic forms ωt such that [ωt] · [V ] > 0. (2) We do not claim that any ω˜−surface in the class A is smoothly isotopic to V . In fact, even for a ﬁxed symplectic structure, there are plenty of non-uniqueness results (see for example [17], [12], [13], [14]). However, we have the following observation. Corollary 3.3. Suppose ω and ω˜ are two deformation equivalent symplectic + forms on a 4-manifold M with b (M) = 1. If A ∈ H2(M, Z) is a homology class pairing positively with both ω and ω˜, then there is a 1-1 correspondence of smooth isotopy classes of connected ω− and ω˜−symplectic surfaces in the class A. As an immediate corollary of this theorem and the light cone lemma we obtain:

Corollary 3.4. In the situation of Theorem 3.1, if Ai ·Ai ≥ 0 for all i ∈ |G|, then V˜ exists for any ω˜ deformation equivalent to ω. If M has Kodaira dimension κ(M) = −∞, then the deformation class of ω is determined by the canonical class Kω (see [38], [21] for rational, [31] for irrational ruled manifolds). This immediately implies the following: Corollary 3.5. In the situation of Theorem 3.1, suppose further that M has κ(M) = −∞. Then V˜ exists for any G−positive ω˜ with Kω˜ = Kω. In particular, if Ai · Ai ≥ 0 for all i ∈ |G|, then V˜ exists for any ω˜ with Kω˜ = Kω. The methods employed to prove Theorem 1.3 are rather robust and allow some variation. The following is an example. Theorem 3.6. Let (M, ω) be a symplectic manifold with b+(M) = 1 and G a homological conﬁguration represented by a curve conﬁguration V . Fur- thermore, let Σ be a connected embedded ω-symplectic curve disjoint from V . Let ω˜ be any symplectic form on M such that the following hold:

(1) ω˜ is deformation equivalent to ω through forms ωt which leave Σ symplectic and (2) ω|Σ =ω ˜|Σ. 18 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

Then there exists a curve conﬁguration V˜ for ω˜ such that V˜ is disjoint from Σ and V˜ is smoothly isotopic (with respect to G) to V . Proof. Theorem 2.9 provides a symplectic form τ deformation equivalent to ω, which makes Σ t V τ-symplectic and with [τ] = [˜ω]. As τ is deformation equivalent to ω, we obtain a family {αt} of symplectic forms from τ toω ˜ which satisfy the following:

(1) [α0] = [τ] = [˜ω] = [α1]; (2) Σ is αt-symplectic and (3) τ|Σ =ω ˜|Σ Here (3) is achieved by Moser’s method on Σ. Then by Thm. 1.2.12, [39], there is a family of symplectic forms αst (s, t ∈ [0, 1]) such that α1t is a cohomologous deformation of τ toω ˜ with α1t|Σ = τ|Σ =ω ˜|Σ = ω|Σ. Now apply the Moser Lemma again to obtain a Hamiltonian isotopy that is identity on Σ, from which we produce V˜ as claimed. ¤

3.2. Existence of Diffeomorphic surfaces. We briefly remark on the consequences of the results in previous section coupled with actions of diffeomorphism groups; notation introduced here will be used throughout the rest of the paper. Recall from (1.1) that D(M) is the image of the group of diffeomorphisms Diff(M) in Aut(H2(M, Z)). D(M) defines a group action on the set of symplectic canonical classes K of M. When M has b+(M) = 1, up to sign, D(M) acts transitively on K (see [32], [31]). For symplectic manifolds with κ(M) = −∞ this result can be improved: Lemma 3.7 ([31], [32]). If M has Kodaira dimension κ(M) = −∞, then the action of D(M) on K is transitive. Furthermore, D(M) is generated by reflections on (−1−) and (−2)-smooth spherical classes. Concretely, these are Ei and H − Ei − Ej − Ek, i 6= j 6= k 6= i for rational manifolds and Ei, F − Ei − Ej, i 6= j for irrational manifolds. This result reduces the problem for symplectic manifolds with κ(M) = −∞ to understanding those classes A ∈ H2(M) admitting symplectic representatives for symplectic forms ω within a fixed symplectic canonical class K ∈ K. Corollary 3.8. Let (M, ω) be a symplectic manifold with κ(M) = −∞ and A ∈ H2(M, Z) a homology class admitting an ω-symplectic surface V . Then for every symplectic canonical class K there exists a symplectic form ω˜ with Kω˜ = K admitting a ω˜-symplectic surface diffeomorphic to V .

Proof. Lemma 3.7 provides for an element of D(M) which takes K to Kω. This element covers a diﬀeomorphism; letω ˜ be the pull-back of ω under this map. The result then follows. ¤ STABILITY AND EXISTENCE OF SURFACES 19

For a general symplectic manifold, one may consider the following subset of Ω(M): Definition 3.9. Let D(ω, A) ⊂ Ω(M) be the set of symplectic forms on M satisfying the following: For every α ∈ D(ω, A) there is a symplectic form β in the Diff(M)−orbit of α which has canonical class Kβ = Kω and [β] · A > 0. Denote by Dd(ω, A) ⊂ D(ω, A) the set of classes such that β is deformation equivalent to ω. Thus D(ω, A) is the orbit under the action of Diff(M) of the set {β ∈ d Ω(M) | Kβ = Kω, [β] · A > 0} whereas D (ω, A) is the orbit of the path connected component of {β ∈ Ω(M) | Kβ = Kω, [β] · A > 0} containing ω. The following results extend the stability results from just the orbit of ω to these larger sets when κ(M) = −∞, as a consequence of Lemma 3.7. Lemma 3.10. Let (M, ω) be a symplectic manifold with κ(M) = −∞. Then D(ω, A) = Dd(ω, A) and the restriction of the map Ω(M) → H2(M, R) ω 7→ Kω to D(ω, A) is onto the set of symplectic canonical classes. Theorem 1.3 can be used to obtain the following general existence prin- ciple for manifolds with b+(M) = 1. Proposition 3.11. Let (M, ω) be a symplectic manifold with b+(M) = 1 and A ∈ H2(M, Z) a homology class admitting an ω-symplectic surface V . Let ω˜ ∈ Dd(ω, A). Then there exists an ω˜−symplectic surface V˜ which is diffeomorphic to V . Proof. Let α be any symplectic form in the Diff(M)-orbit ofω ˜ such that α is deformation equivalent to ω and [α] pairs positively with A. Then by Theorem 1.3 there exists an α-symplectic submanifold Vα smoothly isotopic to V . Combining this with the diffeomorphisms taking α toω ˜, the result follows. ¤ Under the additional assumption that κ(M) = −∞, two symplectic forms with a common canonical class are deformation equivalent [40], hence the above result can be sharpened. Lemma 3.12. Let (M, ω) be a symplectic manifold with κ(M) = −∞ and A ∈ H2(M, Z) a homology class admitting an ω-symplectic surface V . Let ω˜ ∈ D(ω, A). Then there exists an ω˜−symplectic surface V˜ which is diffeomorphic to V . Again the light cone lemma allows us to formulate a simple corollary when κ(M) = −∞. Corollary 3.13. If A · A ≥ 0, then for any symplectic form ω˜, there exists an ω˜−symplectic surface V˜ which is diffeomorphic to V . 20 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

4. Lagrangian ADE-configurations In this section, as an application of the stability results, we explain how to obtain Lagrangian ADE-configurations . This is closely related to the conifold transition, which we will review in Section 4.1, where a slight re- finement of the deformation type result in [41] is shown. A stability result of Lagrangian ADE-configurations will be explained in 4.1. This will even- tually lead to a proof of Corollary 1.4.

4.1. Conifold transitions and stability of Lagrangian configurations. By definition, in real dimension 4, an ADE-configuration of Lagrangian spheres is a plumbing of Lagrangian spheres as An, n ≥ 1; Dn, n ≥ 4; or E6,7,8 Dynkin diagrams. These are the smoothing of simple singularities of type C2/Γ, where Γ is a finite subgroup of SU(2). On the other hand, one may perform a minimal resolution of such singularities, which incurs a tree-like configuration of (−2)-rational curves, which is of the same diffeomorphism type as the Lagrangian plumbings in smoothings. One may replace a neighborhood of a smoothing by the resolution, or vice versa. Such a surgery is called a conifold transition. See [44], [41] for more background on conifold transitions over surfaces. Note that performing the conifold transition as a symplectic cut-and-paste surgery has the following features: S (i) If there are ω and ωe-Lagrangian ADE-configurations L = i{Li} and e S e L = i{Li}, respectively, then one may choose an appropriate neighborhood of N and Ne for L and Le to perform conifold transitions. Symplectically the conifold transitions remove N and Ne and replace them by a neighborhood of a symplectic ADE-configuration of (-2)- spheres. By the Lagrangian neighborhood theorem for configurations (Proposition 7.3 of [44]) such a symplectic configuration can be chosen isomorphic for both surgeries on L and Le. In other words, suppose ω0 and ωe0 are the symplectic forms after conifold transitions, where S S e i{Vi} and i{Vi} are the symplectic configurations, then one may 0 0 choose the surgeries so that ω (Vi) = ωe (Vi) for all i. n (ii) Moreover, let {[Li]}i=1 span a subspace L ⊂ H2(M, R). One may consider its orthogonal complement L⊥ under Poincare pairing. The conifold transition changes the symplectic form, adopting notation from 0 the previous paragraph, in such a way that ω|L⊥ = ω |L⊥ . This applies equally well in the other direction of the transition, that is, when changing a resolution {Vi} to a smoothing. As a consequence, if [ω] = [ωe] and S S e they each admit a symplectic ADE-configuration i{Vi} and i{Vi} so that [Vi] = [Vei], after changing both configurations to smoothings the new symplectic forms are again cohomologous. In our situation, we would like to understand the connection between conifold transition and symplectic deformations. Symplectically, Ohta and STABILITY AND EXISTENCE OF SURFACES 21

Ono showed in [41] that any weak/strong symplectic filling of the link (L, λ) of an ADE-singularity has a unique symplectic deformation type, while the deformation is along a family of weak/strong symplectic fillings. Here λ is the contact form on the link L. We refer readers to [41] (or some standard reference on contact geometry and symplectic fillings, e.g. [15]) for relevant definitions. Because of the local feature of conifold transitions, it is rather conceivable that it can be achieved by a compactly supported symplectic deformation. In particular this is true for an A1 smoothing in view of symplectic cuts. Unfortunately we are unable to prove this: note that this is not a local question, for example, one cannot obtain a compactly supported symplectic deformation in T ∗S2 so that the zero section becomes symplectic while a fiber is preserved as a Lagrangian plane due to homological obstructions. However, we show the following variant of Ohta and Ono’s result. We emphasize in this result that there is no guarantee that ω1 is the symplectic form obtained by conifold transition (as a surgery).

Lemma 4.1. Let (W, ω) be a neighborhood of an ADE-symplectic conﬁgura- tion V which is a strong ﬁlling of (L, λ). Then there is a compactly supported symplectic deformation {ωt}0≤t≤1 on V , so that ω0 = ω and V ⊂ (W, ω1) is a Lagrangian ADE-plumbing. The reverse procedure also exists, that is, one has a compactly supported deformation which transforms smoothings into resolutions.

Proof. We only prove the direction from resolution to smoothings, the other direction is identical. First perform a conifold transition to V which incurs a symplectic manifold (W,f ωe) diffeomorphic to W with a smoothing configuration Ve. One identifies W and Wf smoothly so that Ve is identified to V , hence the result is a symplectic form ω0 on W so that V is a ω0-Lagrangian configuration and ω = ω0 near ∂W since conifold transition only happens in the interior. From [41], we have a deformation {Ωt} which is a symplectic deformation 0 from ω to ω , where Ωt are all strong fillings of (L, λ). By definition, this means in a collar neighborhood U of L, with U ∩ V = ∅,Ωt = dλt, and λ is an extension of λ on L. Take Xe so that i Ω = d λ . Cut off Xe t t Xet t dt t t so that one obtains Xt which is supported in U and equals Xet in a smaller U 0 ⊂ U. Note that the right hand side vanishes identically on L, the flow of Xt is supported away from L and creates a family of diffeomorphisms 0 ϕt such that (ϕt)∗Ωt = ω in U . Hence {(ϕt)∗(Ωt)}0≤t≤1 is a compactly supported deformation of ω, while ω1 = (ϕ1)∗(Ω1) contains a Lagrangian configuration, since (ϕt)∗Ωt = Ωt in the complement of U ¤

With this understood, we may show: 22 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

Theorem 4.2. ADE-conﬁgurations of Lagrangian spheres have the stability property in symplectic manifolds M with b+(M) = 1. Moreover, if D ⊂ M is a smooth symplectic divisor, then the stability holds in its complement.

Proof. We give the proof in the presence of D, the case when D is empty is only easier. Given a symplectic manifold (M, ω), suppose it has an ADE- Sn configuration L = i=1{Li} consisting of Lagrangian spheres Li in M\D. Consider ωe deformation equivalent to ω through a compactly supported deformation family in M\D, where ωe([Li]) = 0. We would like to show that there exists an ADE-plumbing of Lagrangian spheres in the complement of D. We proceed as follows. Apply first Lemma 4.1 to a neighborhood N of L, which turns it into a resolution. This results in a new symplectic form 0 S ω , as well as a symplectic configuration i Vi. Note that by choosing N sufficiently small, one may assume ω and ω0 are C0-close, which is equivalent 0 to saying ω |L being small. 0 When ² = ||ω − ω ||C0 is sufficiently small, one allows a symplectic 0 0 0 0 deformation from ωe to ωe , so that [ωe ]|L = [ω ]|L,[ωe ]|L⊥ = [ωe]|L⊥ and 0 ² > ||ωe − ωe ||C0 . This can be achieved by packing-blowup correspondence [40] for the following reason. Both L and L⊥ are spanned by subcollections in {H,E1,...,En}, while one has the freedom to adjust the symplectic areas of each: sizes of ball-packings corresponding to symplectic areas of Ei which can be adjusted slightly by the continuity of packing, while the area of H can be adjusted by a global rescaling. Also, note that when ² is sufficiently small, D is preserved as a symplectic divisor. Now apply the Stability Theorem 3.6 for the symplectic configuration S 0 0 i Vi from ω to ωe and divisor D as Σ. This implies the existence of a symplectic configuration with respect to ωe0 in the complement of D. One can then use Lemma 4.1 in a reverse direction on this configuration to obtain a smoothing (Lagrangian configuration of spheres) in M with a certain symplectic form ωe00. Note that ωe and ωe00 are deformation equivalent by concatenating the symplectic deformation from ωe to ωe0, and they are cohomologous by (i) and (ii) (because this reverse conifold transition only “erases” the symplectic form on L and leaves L⊥ invariant). Applying Thm. 1.2.12, [39], one may deform such a symplectic defor- 00 mation to an isotopy of symplectic forms Ωt,Ω0 = ωe and Ω1 = ωe by symplectic inflations while preserving D as a symplectic divisor. Along this isotopy of symplectic forms, D has constant symplectic area. Therefore, one ∗ may choose a diffeomorphism τt supported near D, so that τt (Ωt) is con- ∗ stant on D. Now Moser’s method on τt (Ωt) yields an isotopy φt which is ∗ identity restricted to D, where (φ ◦ τ1) (Ω1) = ωe. Then the φ1-image of the constructed ωe00-Lagrangian configuration is as desired in the complement of D. ¤ STABILITY AND EXISTENCE OF SURFACES 23

4.2. Existence. In [34], the second and third authors derived a necessary 2 2 and sufficient condition for A ∈ H2(CP #kCP , Z) to admit a Lagrangian spherical representative: this holds if and only if A is D(M)-equivalent to E1 − E2 or H − E1 − E2 − E3 and [ω] · A = 0. With the stability result above, we may improve the existence part into existence of ADE-smoothings. [41] explained how to compactify an ADE-type smoothing into a rational manifold of diffeomorphism type CP2#(n + 3)CP2. After compactification, a symplectic neighborhood of the Lagrangian configuration can be recovered by removing a set of smooth symplectic divisors from the rational surface. The homology classes of these divisors are listed as follows:

• An : H,H − E1 − · · · − En+1; • Dn : E1,E2 − E1,H − E1 − E2 − E3, 2H − E1 − E2 − E4 − · · · − En+3; • En : E1,E2 −E1,E3 −E2 −E1, 3H −2E3 −E4 −· · ·−E9(−E10 −E11). Note that for the case of An we have used a particularly simple set of divisors slightly diﬀerent from that in [41], where we have CP2#(n + 1)CP2 as the ambient rational surface. The corresponding homology classes of the Lagrangian ADE-conﬁgurations are given as follows:

• • • • E −E An : E1−E2 E2−E3 ··· En−1−En n n+1

E −E Dn : 4 • 5 • • • • E −E −H456 E6−E4 E7−E6 ··· n+3 n+2

E −E E6(7,8) : 4 • 7

2 • • • • • ( • • ) −H4−9 H H479 E6−E7 E5−E6 E8−E5 89(10) E10−E11

2 Here Hijk and H4−9 are shorthand for H − Ei − Ej − Ek and 2H − E4 − · · · − E9, respectively. Motivated by these explicit identiﬁcations, we deﬁne:

n 2 2 Definition 4.3. A set of homology classes {li}i=1 ⊂ H2(CP #nCP , Z) is a homological Lagrangian ADE-configuration if there is a D(M)- 2 2 n equivalence τ on H2(CP #kCP ) so that {τ(li)}i=1 are of the form specified above and ω(li) = 0. We are now ready to prove Corollary 1.4, which we recall below. Corollary 4.4. In rational or ruled 4-manifolds, any homological Lagrangian n ADE-configuration {li}i=1 admits a Lagrangian ADE-configuration representative. In the case of An-configurations, one may require the configuration lie in M\D, where D is a symplectic divisor disjoint from a set of n+1 embedded symplectic representatives of {Ei}i=1 . 24 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

Proof. Notice first that any reflection along a −2-sphere is the homological action of a diffeomorphism, therefore we may assume the homological configuration is precisely of the form specified in [41]. Choose an arbitrary symplectic form ω0 so that our designated classes admit a symplectic ADE- configuration representative (see [41] for an example). Then one may obtain a Lagrangian ADE-configuration by a conifold transition, by changing the symplectic form to some ω00. Note that ω and ω00 are symplectic deformation equivalent (as is the case for any symplectic form in rational manifolds with the same canonical class), our existence result is an immediate consequence of Theorem 4.2. For the An case, we refine our embedding of the Lagrangian configuration as follows. Blow down all Ei, i ≤ n + 1 and shrink the resulting balls to a very small equal size, then isotope them into a Darboux neighborhood. Upon blowing back up these small balls one obtains a symplectic form on M with an open set symplectomorphic to B4#(n+1)CP2, where all exceptional spheres have the same symplectic area. This open set contains a Lagrangian An-configuration, see for example, the construction in Section 2 of [52]. While the isotopy above can be chosen disjoint from D, the deformation is supported disjoint from D, as well. Therefore, one may apply the stability result in Theorem 4.2 above. ¤

Proof of Corollary 1.5 and 1.6. Note that we may reduce Corollary 1.5 to the case of 1.6, that is, when the packing is supported away from the isotropic skeleton. To see this, rescale the symplectic form on M so that ω(Ei) are rational numbers. Then choose a deformation so that the symplectic form of the minimal model of M has rational period, this can be done due to the openness of the non-degeneracy condition. One then shrinks all embedded balls corresponding to each exceptional sphere (including those not listed as Ei, but consisting basis elements in H2(M)) to a very small volume and then move them away from an isotropic skeleton of the minimal model of M. The blow-up along such small balls thus gives a form ω0 which is deformation equivalent to the original symplectic form. If one has an An- Lagrangian configuration for ω0, then the stability for manifolds with b+ = 1 in Theorem 4.2 concludes Corollary 1.5. Therefore, it suffices to find a Lagrangian An-configuration in the complement of the isotropic skeleton when the minimal model of M has rational period (case of Corollary 1.6). Biran [5, Theorem 1.A] showed that this complement is symplectomorphic to a standard symplectic disk bundle E modelled on the normal bundle of a Donaldson hypersurface. One may then compactify this disk bundle E into a symplectic ruled surface E0 by slightly deforming the symplectic form and adding a symplectic divisor at infinity (equivalently, do a symplectic cut near the boundary). Upon blowing up, STABILITY AND EXISTENCE OF SURFACES 25 one may apply the existence result Corollary 1.4 with D as the added infinity divisor. The corollary is thus concluded by embedding the complement of D back into the complement of the isotropic skeleton. ¤

5. Spheres in Rational Manifolds In this section, we prove the classification result Theorem 1.8, using the following strategy: • we first provide a classification of homology classes that satisfy the imposed constraints, see Section 5.1; • secondly, we show that all classes obtained in this way are symplectically representable by a connected ω-symplectic sphere for some symplectic structure ω, using the so-called tilted transport of Section 5.2; • finally, we apply the results in Section 3 to extend the result to all classes satisfying conditions (1)-(3) in Spec. 1.10. In Section 5.3 we also include a complete account for symplectic −1, −2, −3- spheres for completeness; these results mostly follow from earlier work, see [29, 32, 34].

5.1. Homology classes of smooth −4 spheres. Consider a class A ∈ 2 2 H2(M, Z) for M = CP #kCP . In the standard basis we write A = aH − Pk i=1 biEi. Such a class is called reduced if • b1 ≥ b2 ≥ ... ≥ bk ≥ 0 and • a ≥ b1 + b2 + b3. The following lemma gives a complete list of non-reduced classes for smooth −4 spheres.

2 2 Lemma 5.1. Let M = CP #kCP with k ≥ 1 and A ∈ H2(M, Z). Assume that A · A = −4. Then, up to D(M)-equivalence, A is a reduced class (when k ≥ 3), or one of the following:

−H + 2E1 − E2, 2E1, 2(H − E1 − E2),H − E1 − .. − E5. 2 2 2 Proof. For k = 1, (aH − bE1) = a − b = −4 implies that only ±2E1 is possible. For k = 2, Lemma 1, [27] reduces the problem to classes with 2a ≤ b1 +b2. 2 2 Thus 3b1−2b1b2+3b2 ≤ 16 and the only possible classes are D(M)-equivalent to 2E1, 2(H − E1 − E2) and −H + 2E1 − E2. For k ≥ 3, as in Lemma 3.4, [26], it can be shown using reﬂections along −2-spheres H − Ei − Ej − Ek that either A is D(M)-equivalent to a reduced class or to one that satisﬁes 3 b2 + b2 + b2 − 4 ≤ a2 ≤ (b2 + b2 + b2). 1 2 3 4 1 2 3 26 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

2 P 2 In addition to this inequality, a − bi = −4 and b1 ≥ b2 ≥ ... ≥ bk ≥ 0. The solutions to this system, written in short as (a, b1, b2, .., ), are: (0, 1, 1, 1, 1, 0..), (1, 1, 1, 1, 1, 1, 0, ..), (0, 2, 0, ..), (1, 2, 1, 0, ..), (2, 2, 1, 1, 1, 1, 0, ..), (2, 2, 2, 0, ..), (3, 2, 2, 2, 1, 0, ..), (3, 3, 2, 0, ..). Under the D(M)-action, (0, 2, 0, ..) ↔ (2, 2, 2, 0, ..), (1, 1, 1, 1, 1, 1, 0, ..) is in a class of its own when k = 5. The other classes are all equivalent, and when k ≥ 6, (1, 1, 1, 1, 1, 1, 0, ..) is included as well. ¤

5.1.1. Symplectic Genus. In order to address the reduced classes in Lemma 5.1, we first briefly describe a general obstruction to the existence of smooth /symplectic surfaces in a symplectic manifold. Clearly, for a class A ∈ H2(M, Z) of a symplectic manifold (M, ω) to be represented by a symplectic surface, there must exist α ∈ CM with α·A > 0. Let K denote the set of symplectic canonical classes. Consider the following set: KA = {K ∈ K | ∃α ∈ CM : Kα = K, α · A > 0}. 1 To each K ∈ KA, define ηK (A) = 2 (K · A + A · A) + 1. Finally, define the symplectic genus to be η(A) = max ηK (A). K∈KA

Note that there is no guarantee that η(A) ≥ 0. If K ∈ KA is some symplectic canonical class such that η(A) = ηK (A), we obtain the inequality K˜ · A ≤ K · A for any K˜ ∈ KA. Moreover, Lemma 3.2, [26] shows that η(A) has the following properties: (1) The symplectic genus η(A) is no larger than the minimal genus of A. Moreover, if A is represented by a connected symplectic surface, then the minimal genus and the symplectic genus coincide. (2) The symplectic genus is invariant under the action of Diff(M). Notice that the first condition ensures that the symplectic genus is well- defined as well as providing an obstruction to the existence of a smooth / symplectic curve. For reduced classes A in non-minimal rational or ruled manifolds, Lemma 3.4, [26], proves that Kst ∈ KA. Thus we obtain the following: Lemma 5.2. Let M be a non-minimal rational or ruled manifold and A ∈ H2(M, Z). Assume that A is reduced and A can be represented by a smooth 2 sphere. Then Kst · A ≤ −2 − A .

2 2 Example. The class e = (11, 6, 6, 6, 1, .., 1) ∈ H2(CP #18CP , Z) satis- ﬁes e · e = −4 and the adjunction equality for an embedded sphere for some K ∈ K. However, it is D(M)-equivalent to the reduced class er = (4, 1, .., 1), STABILITY AND EXISTENCE OF SURFACES 27 which has symplectic genus 1, hence cannot be represented by a smooth embedded sphere. It then follows from stability that the same must hold for e. Note also that e is not Cremona equivalent (reﬂections with respect to only (-2)-spherical classes) to a reduced class, this consideration distinguishes this from the approach of [34].

5.1.2. Reduced −4 classes. We will now begin a study of the possible reduced 2 2 classes. Let M = CP #kCP with k ≥ 1 and A ∈ H2(M, Z). Assume that A is reduced, A · A = −4 and A can be represented by a smooth sphere. Thus Lemma 5.2 implies that Kst ·A ≤ 2. Concretely, for some d ∈ Z, d ≤ 2 k and τ ≥ 0, the array of coeﬃcients (a, b1, b2, ..., bk) ∈ Z of such a class A solves:

Xk (5.1) 3a = bi − d i=1 Xk 2 2 a = bi − 4(5.2) i=1 (5.3) a ≥ b1 + b2 + b3

(5.4) bi ≥ bi+1 ≥ τ, i = 1, . . . , k − 1.

Notice that for a class to have negative self-intersection and be reduced, we must have k ≥ 10. The role of τ will become transparent in the proof, we will only consider cases with τ ∈ {1, 2, 3} (the resulting equations are in fact not exclusive). In summary, the standing assumptions for the set of equations (5.1)-(5.4) are:

(5.5) d ≤ 2, d ∈ Z, k ≥ 10, and τ ∈ {1, 2, 3}.

The goal is now to show that d = Kst · A = 2 is the only possibility, even k+1 when we relax the condition to allow (a, b1, b2, ..., bk) ∈ R . For this we need to describe a rearrangment operation which will allow us to rule out these cases.

k+1 Lemma 5.3. Assume a solution (a, b1, . . . , bk) ∈ R to (5.1)-(5.4) exists when either • τ = 1 and k ≥ 11 or • τ ∈ {2, 3} and k = 10 or • τ = 1, d < 2 and k = 10. 0 0 k+1 Then there exists a solution (a, b1, ..., bk) ∈ R to (5.1)-(5.4) which further satisﬁes: 28 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

0 0 0 (5.6) a = b1 + b2 + b3 0 0 (5.7) b2 = ··· = b4+r−1 = B 0 0 (5.8) b4+r+1 = ··· = bk = τ 0 0 (5.9) τ ≤ b4+r = b ≤ B where 0 ≤ r ≤ k − 4. Proof. We ﬁrst describe a rearrangement operation on a solution to (5.1)- (5.4) which changes the bi while leaving a unchanged and preserving all but (5.2). Suppose s = (a, b1, . . . , bk) is a solution to (5.1)-(5.4). Assume + bi > bj > 0. Then for c ∈ R , replace (bi, bj) by (bi+c, bj −c). This operation clearly leaves (5.1) unchanged and, by properly choosing c, preserves (5.3) and (5.4). We will always assume that c has been chosen in this manner. 2 2 2 After such an operation, bi + bj will increase at least by 2c . Now apply this operation repeatedly choosing bi ∈ {b1, b2, b3} and bj ∈ {b4, .., bk}. One arrives at one of the following scenarios:

• a = ˜b1 + ˜b2 + ˜b3 or ˜ ˜ ˜ ˜ ˜ • a > b1 + b2 + b3 and b4 = ··· = bk = τ. In the ﬁrst case, one then further rearranges ˜b1 with ˜b2 until ˜b2 = ˜b3. ˜ ˜ ˜ ˜ ˜ Then rearrange b4 with the rest until b4 = b3 or b5 = ··· = bk = τ. If ˜b4 = ˜b3, do further rearrangements so that ˜b5 = ˜b4, etc. In the second case, ˜ ˜ ˜ ˜ ˜ rearrange b2 and b3 with b1 until b2 = ··· = bk = τ. The end result is a new sequence

0 0 0 s = (a, b1, . . . , bk) that satisfies (5.1), (5.6)-(5.9) for some 0 ≤ r ≤ k − 4 as well as one of the following: 0 0 0 (1) a = b1 + b2 + b3 or 0 0 0 (2) a > b1 + 2τ and b2 = ··· = bk = τ. Notice that s0 will not necessarily satisfy (5.2), instead one has X 2 02 (5.10) a ≤ bi − 4 0 0 0 2 2 2 Case 1: Consider a = b1+b2+b3. The function F (s) = a −b1−· · ·−bk +4 satisfies F (s0) ≤ 0. 00 k−7 d 00 00 00 00 00 Let b1 = 2 τ − 2 , a = b1 + 2τ and s = (a , b1, τ, .., τ). Then in all cases to be considered we have 00 00 b1 ≥ τ and F (s ) ≥ 0. Thus s00 satisfies (5.1), (5.6)-(5.9) just as s0 does. Therefore, the line segment between s00 and s0 in Rk+1 must contain a solution to F (s) = 0. Moreover, such a solution must satisfy (5.1) and STABILITY AND EXISTENCE OF SURFACES 29

(5.6)-(5.9) since all these conditions are convex and the endpoints of the chosen segment satisfy all these restrictions. 0 Case 2: Consider the situation that we obtain a solution with a > b1 +2τ 0 0 0 and b2 = ··· = bk = τ. By solving 3a = b1 + (k − 1)τ − d for (k − 1)τ and 0 substituting this into (5.10) and making use of a > b1 + 2τ, we obtain ³ τ ´2 ³ τ ´2 (5.11) b0 + < b0 − + 2τ 2 + dτ − 4 1 2 1 2 0 When τ = 1 (independent of k in fact), (5.11) admits no solution b1 ≥ 1 when d ≤ 2. Assume now that τ ∈ {2, 3} and k = 10. Then (5.11) simplifies to d b τ < τ 2 + τ − 2. 1 2 d Assume that τ = 2. Then 2 ≤ b1 < 1 + 2 ≤ 2, which is a contradiction. d 2 Assume that τ = 3. Then 3 ≤ b1 < 3 + 2 − 3 which has no solution when d ≤ 1. For d = 2, consider again (5.1). Solving for b1 under the assumption 0 5 a > b1 + 2τ, one obtains 3 ≤ b1 < 2 , also a contradiction. Hence Case 2 never shows up and the proof is completed. ¤ Remark 5.4. Consider d ≤ 1 and replace 4 by 3 in (5.2) to consider classes with A · A = −3. Then Lemma 5.3 continues to hold for τ = 1 and k ≥ 10. This can easily be seen in Case 2, where in (5.11) the final term changes to 3. Moreover, in Case 1 the same point s00 can be used. It should be noted that in Case 1, the case τ = 1, d = 2 and k = 10 does not work. It can be shown that in this setting the procedure will not terminate with a solution as described. The reason for this becomes clear when one considers Lemmata 5.6 and 5.7. Making use of this process, we now begin to rule out certain reduced classes. Proposition 5.5. Let M = CP 2#kCP 2. Then there exists no reduced class A ∈ H2(M, Z) with A · A = −4 and that min{bi} ≥ 1 in the following cases: (1) k ≥ 11 and Kst · A ≤ 2; (2) k = 10 and Kst · A < 2; (3) k = 10, Kst · A = 2 and min{bi} ≥ 3. Before we pass to the proof, let us briefly consider the ramifications of this result. Recalling Lemma 5.1, this result shows that when k ≥ 11, we have no reduced classes with A · A = −4 which can be represented by a smooth sphere. Moreover, according to this result, when k = 10, if A is to be represented by a smooth sphere, then A must satisfy Kst · A = 2 and b10 ∈ {1, 2}. The latter cases will be considered after the proof. Proof. We will proceed to show that there exists no solution to (5.1), (5.2) and (5.6)-(5.9) under the conditions given in the theorem. The three cases correspond to 30 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

(1) k ≥ 11, d ≤ 2 and τ = 1; (2) k = 10, d < 2 and τ = 1; (3) k = 10, d = 2 and τ = 3. To simplify notation, drop all ’ in (5.6)-(5.9). Consider ﬁrst the case τ = 1 and k ≥ 11. Then as a = b1 + 2B, we obtain

(5.12) 2b1 = (r − 4)B + b + k − 4 − r − d from (5.1) and using this in (5.2) it follows that (5.13) 0 = (r − 6)B2 + 2Bb − b2 + (k − 4 − r − d)(2B − 1) + 4 − d. Notice that 2Bb − b2 and 4 − d are strictly positive. 6 ≤ r ≤ k − 6: In this case, (r − 6)B2 and (k − 4 − r − d)(2B − 1) are non-negative, hence no solution exists. As k ≥ 11, we need to consider k = 11 and r = 6 separately: (5.13) reduces to 0 = 2Bb − b2 − 2B + (2 − d)2B + 3 = (B − 1)2 − (B − b)2 + (2 − d)2B + 2 from which it can be seen that no solution exists if b ≥ 1. r = k − 5 > 6: When r = k − 5 > 6, then (5.13) reduces to 0 = (k − 12)B2 + 2Bb − b2 + (B − 1)2 + (2 − d)2B + 2 which again has no solution. r = k − 4: (5.13) can be rewritten as 0 = (k − 11)B2 + 2Bb − b2 + (B − 2)2 + 3 which admits no solution as B ≥ b ≥ 1. For the following cases, determine

k − 4 − r − d = 2b1 − (r − 4)B − b and insert into (5.13) to obtain 2 2 (5.14) 0 = (2 − r)B + 4Bb1 − b − 2b1 + (r − 4)B + b + 4 − d

Note that 2b1 − (r − 4)B − b ≥ 0, and thus if d = 2 and r = 5 we must have k ≥ 11. This is the cause for the restriction to d < 2 in the case k = 10 and τ = 1. Therefore, all of the following arguments continue to hold when k = 10, d < 2 and τ = 1. r = 0: In this case (5.14) becomes 2 2 0 = 2B + 4b1B − b − 2b1 − 4B + b + 4 − d = 2 2 2 = B − b + 2Bb1 − 2b1 + (B − 2) + 2Bb1 − d + b which admits no solution. r = 1: As before, (5.14) becomes 2 2 0 = B − b + 2Bb1 − 2b1 + 2Bb1 − 3B + b + 2 + 2 − d 1 where 2Bb1 − 3B + b + 2 ≥ 2 . Hence no solution exists. r = 2 : Again (5.14) becomes 2 0 = 4Bb1 − b − 2b1 − 2B + b + 4 − d STABILITY AND EXISTENCE OF SURFACES 31 which can be rewritten to show that no solution exists here either. r = 3, 4 : Write b1 = B + α and insert into (5.14). Then again it can be shown that no solution exists. r = 5 : Again write b1 = B + α and insert into (5.14). When α ≥ 2 1 it easily follows that there exists no solution. Otherwise |b1 − B| < 2 and using this in (5.14) to succesively estimate the diﬀerences of the B and b1 terms it can again be shown that no solution exists. This completes the case with τ = 1 and k ≥ 11. As noted before, the cases with 0 ≤ r ≤ 5, k = 10, d < 2 and τ = 1 have also been completed. It remains to consider r = 6 in this setting. r = 6, d < 2 and k = 10: (5.13) reduces to 0 = 2Bb − b2 − 2dB + 4 = (B − 1)2 − (B − b)2 + 2B(1 − d) + 3 which admits no solution when d ≤ 1. We now turn to τ = 3, k = 10 and d = 2. Rewriting (5.1), (5.2) and (5.6)-(5.9) it follows that

(5.15) [(r − 5)B2 + (32 − 6r)B + 9r − 50] − (B − b0)2 = 0;

0 (5.16) 2b1 = (r − 4)B + b + 3(6 − r) − 2. One can then check the compatibility of these equations, case-by-case, for b0+2 0 r from 0 to 6. For r = 6 one easily shows explicitly B = 2 < b from the 0 0 ﬁrst equation. For r = 5, from (5.16), 2b1 = B + b + 1 so B − b ≤ 1. This again contradicts (5.15). 0 For r ≤ 4, from (5.16) one deduces 2b1 ≤ (r − 3)b + 16 − 3r, implying 7 b1 ≤ 4 when r = 4 and b1 ≤ 2 when r ≤ 3. The minimum of the B-quadratic 7 expression in (5.15) is taken at B = 3 when 1 ≤ r ≤ 4 and B = 2 for r = 0. 0 2 1 Also −(B − b ) ≥ −1 for r = 4 and ≥ − 4 for r ≤ 3. Each case will imply a positive minimum in (5.15), which concludes our proof. ¤ As noted before, when k = 10 this result implies that A must satisfy Kst · A = 2 and b10 ∈ {1, 2}. We now show that if b10 = 1 we obtain no solutions either. Lemma 5.6. Let M = CP 2#10CP 2. Then there exists no reduced class A ∈ H2(M, Z) with A · A = −4, Kst · A ≤ 2 and b10 = 1.

Proof. Assume such a class exists. If A = (a, b1, .., b9, 1), then the class 2 2 (a, b1, .., b9) in H2(CP #9CP , Z) is reduced and has square −3. However, an easy computation shows every reduced class in CP 2#9CP 2 has non- negative square. Hence no such class exists. ¤

2 2 Lemma 5.7. Assume that a reduced class A ∈ H2(CP #10CP , Z) satisﬁes A·A = −4 and Kst ·A = 2 with the additional restriction that b10 = 2. Then P9 A = −a(−3H + i=1 Ei) − 2E10 for some a ∈ N and a ≥ 2. 32 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

10 Proof. Then the tuple (a, b1, .., b9) ∈ Z satsiﬁes X (5.17) 3a = bi X 2 2 (5.18) a = bi

(5.19) a ≥ b1 + b2 + b3

bi ≥ bi+1 ≥ 2(5.20)

2 2 hence deﬁnes a reduced class A9 ∈ H2(CP #9CP , Z). Using the formula given for fA9 (to determine the minimal genus) in [24], we obtain fA9 = 1. P9 Theorem 2, [24], thus implies that A9 = −a(−3H + i=1 Ei) for some a ∈ N and a ≥ 2. Therefore A = A9 − 2E10 as claimed. ¤

5.1.3. Classification. Together the results in this section lead to the following Theorem, which completes the smooth classification of Theorem 1.8, as well as implies the exclusiveness part of the symplectic classification of Theorem 1.8: Theorem 5.8. Let M = CP 2#kCP 2 with k ≥ 1 be a symplectic rational surface, and A ∈ H2(M, Z) with A · A = −4. Then A is represented by a smooth sphere if and only if A is D(M)-equivalent to one class in the following list

(1) −H + 2E1 − E2 (2) H − E1 − .. − E5 P9 (3) −a(−3H + i=1 Ei) − 2E10 for some a ∈ N and a ≥ 2 (4) 2E1 (5) 2(H − E1 − E2) Moreover, for A in (1), (2), (3), Kst · A = 2, and there is a symplectic form ω with Kω = Kst such that A · [ω] > 0; there is no symplectic form τ with canonical class Kτ satisfying [τ] · A > 0 and Kτ · A = 2 for classes A of the form in (4), (5). In particular, classes of type (4) and (5) cannot be represented by embedded symplectic spheres for any symplectic form. Proof. Assuming that A is represented by a smooth sphere, Lemma 5.1 gives all the classes in the list except (3). This last class follows from the results of Section 5.1.2, and all other possibilities are excluded. The class H − E1 − .. − E5 can clearly be represented by an embedded symplectic sphere for some symplectic form ω with Kω = Kst. The class −H + 2E1 − E2 can be viewed as the blow-up of a section in a Hirzebruch surface. P9 We will show in Section 5.2 that −a(−3H + i=1 Ei) − 2E10 can be represented by symplectic spheres for some symplectic forms, hence also has smooth representatives. This is a slight overkill: a smooth representative of this class could be constructed directly by the circle sum construction in [28]. We leave that for interested readers. STABILITY AND EXISTENCE OF SURFACES 33

Kst ·A = 2 is clear in (1),(2) and (3).P Choosing 0 < ²i << 1 appropriately, a symplectic form in the class aH − ²iEi has the standard canonical class and pairs positively with A in these cases. Now we analyze the classes of type (4) and (5). The class 2E1 is smoothly representable by a sphere: Consider a smooth sphere in the class E1. A small pushoﬀ of this sphere produces a second exceptional sphere in the same class intersecting once, a smoothing of this will produce a smooth sphere in the class 2E1. Notice that Kτ · 2E1 = 2 implies that −E1 can be represented as a τ- symplectic sphere from Theorem A of [31]. But this means [τ] · (2E1) < 0. The argument is valid for 2(H − E1 − E2), by noticing that H − E1 − E2 is also an exceptional class. ¤

Remark 5.9. For completeness, we describe theP explicit algorithm produc- ing necessary D(M)-equivalences for A = aH − biEi throughout this section, regardless of the value of its square. With such an algorithm, one may determine in a finite number of steps whether a given homology class is represented by a smooth or symplectic sphere given the theorems proven here. This procedure is implicit in [26] and the proof of Lemma 5.1. (1) If a < 0, just change it to −a using reflection along the +1 sphere H. (2) If bi < 0, change it to −bi using reflection along the −1 sphere Ei. (3) Arrange bi ≥ bi+1 using reflections along Ei − Ei+1. (4) Reflect along H − E1 − E2 − E3. 2 (5) Repeat the above process until one arrives at k = 2, k = 3 or b1 + 2 2 2 3 2 2 2 b2 + b3 − 3 ≤ a ≤ 4 (b1 + b2 + b3). Proceed as in Lemma 5.1. 5.2. Tilted transport: constructing symplectic (-4)-spheres.

5.2.1. Reduction from Theorem 1.3. We now consider the existence part of Theorem 1.8. From the assumptions we may assume A has the form specified as type (1),(2) or (3) in the list of Theorem 5.8. We consider type (1). By applying an appropriate diffeomorphism we assume A = −H + 2E1 − E2. Since D(M) acts transitively on the set of symplecticP canonical classes (Lemma 3.7), they are all of the form ±3H + ±Ei. AnyP canonical class with Kω · A = 2 must have Kω = −3H + E1 + E2 + i≥3 ±Ei. Applying trivial transforms on Ei for i ≥ 3 will not affect the pairing Kω · A or ω · A. Hence we may assume Kω = Kst. Any symplectic form with canonical class Kst are deformation equivalent by [40], hence it suffices to construct a symplectic sphere for some symplectic form associated to Kst when A is precisely the class −H + 2E1 − E2. For type (2) we again assume A = H −E1 −· · ·−E5. In this casePKω may be one of the following: it either equals −3H + E + ··· + E + ±E , P 1 5 i≥5 i or 3H − E1 − E2 − E3 + E4 + E5 + i≥5 ±Ei up to reordering the first five exceptional classes. In the latter case, we apply further trivial transforms on 34 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

0 H and E1,2,3 so that A is transformed into A = −H +E1 +E2 +E3 −E4 −E5. 0 However, A is again equivalent to A by reflection along H − E1 − E2 − E3, which does not change the canonical class. The conclusion is A can always be assumed to have the form H − E1 − · · · − E5 while the canonical class can be assumed to be Kst simultaneously. A similar reduction holds for type (3) and we give only a sketch: when P9 δ A = −a(−3H + i=1 Ei) − 2E10 and Kω · A = 2, then Kω = (−1) (−3H + P9 P i=1 Ei) + E10 + i≥11 ±Ei. When δ = 0, the reduction works exactly as in the type (1) case. When δ = 1, again from Theorem A of [31], −H and −Ej for j ≤ 9 and E10 are all represented by ω-symplectic spheres. This implies ω(A) < 0 thus excluded by our assumption. To summarize our discussion, we have the following reduction of Theorem 1.8: Lemma 5.10. Assume A equals any one of the classes of type (1), (2) or (3) specified in Theorem 1.8. If A is represented by a symplectic sphere for some symplectic form ω with Kω = Kst, then Theorem 1.8 holds. We would like to emphasize that A is assumed to equal the classes in Theorem 1.8 instead of being only D(M)-equivalent. Also recall that when Kω = Kst, the symplectic manifold can be assumed to be obtained by blow- ups of symplectic CP2. It is not difficult to verify Lemma 5.10 for classes −H + 2E1 − E2 and H −E1 −..−E5: one may choose a symplectic form ω where ω(Ei) are small enough, then the former class has representatives as iterated blow-ups from an H-sphere. For the latter class, by a change of basis, they are the class F − 2S in S2 × S2#(k − 1)CP2 which clearly has a symplectic representative (here F and S denotes the fiber and base homology classes in S2 × S2). Therefore, the following lemma implies Theorem 1.8: P9 Lemma 5.11. The class −a(−3H + i=1 Ei) − 2E10 has an ω-symplectic representative for some ω with Kω = Kst. The proof of this lemma will occupy the rest of this section.

5.2.2. The Tilted Transport. We start our discussion in a more general con- 2n 2 text. Let π :(E , ωE) → D be a symplectic Lefschetz fibration. This means: −1 2 • (E, ωE) is a symplectic manifold with boundary π (∂D ); 2 • π has finitely many critical points p0, . . . , pn away from ∂D , while π−1(b) is a closed symplectic manifold symplectomorphic to (X, ω) when b 6= π(pi) for any i. • Fix a complex structure j on D2. There is another complex structure Ji defined near pi, so that π is (J, j)-holomorphic in a holomorphic chart (z1, . . . , zn) near pi, and under this chart, π has a local expres- 2 2 sion (z1, . . . , zn) 7→ z1 + ··· + zn. STABILITY AND EXISTENCE OF SURFACES 35

2 Take a regular value of π, b0 ∈ D as the base point. Suppose one has 2r−1 −1 a submanifold Z ⊂ π (b0). We say Z has isotropic dimension 1 if at ⊥ω each x ∈ Z,(TxZ) ∩ TxZ = Rhvxi. We call vx an isotropic vector at x. A simple example of a submanifold of isotropic dimension 1 is a contact type hypersurface. A special case more relevant to us is a closed curve on a surface. Suppose we have a (based) Lefschetz fibration (E, π, b0) with a subman- −1 2 ifold Z ⊂ π (b0) of isotropic dimension 1. Let γ(t) ⊂ D be a path with γ(0) = b0. Assume γ(t) 6= π(pi) for all t and i. Notice there is a natural symplectic connection on E in the complement of singular points as a dis- `n v ⊥ω tribution: for x ∈ E\ i=1{pi}, the connection at x is defined by (T E)x . Here T vE is the subbundle of TE defined by vertical tangent spaces −1 −1 T (π (π(x))) at point x. E|γ = π (γ) thus inherits this connection and thus a trivialization by parallel transports. The symplectic connection also 0 −1 0 defines a unique lift of γ to a vector field of E|γ. We will use π (γ ) to represent this lift. Now choose a vector field V on E|γ tangent to the fibers; one obtains a flow defined by V + π−1(γ0). Suppose the following holds:

Condition 5.12.

• Zt ⊂ Eγ(t) is the time t-ﬂow of Z0 = Z, and each Zt is of isotropic dimension 1.

• For any xt ∈ Zt, let vxt be the isotropic vector. Then ω(V, vxt ) 6= 0. ` Let Zb = Zt. It is then easy to see that Zb is a symplectic submanifold b of E with boundary on Eγ(0) and Eγ(1). We call Z a tilted transport of Z. A special case is when γ(0) = γ(1) and Z0 = Z1. In such cases, one could be able to adjust V appropriately so that Zb is a smooth closed symplectic submanifold, which we will call a tilted matching cycle. In the following we bγ γ sometimes write Z and Zt to emphasize the dependence on the path γ.

Remark 5.13. One notices that the symplectic isotopy class of the tilted transport is independent of speciﬁc choices inside an isotopy classes of the auxiliary data (γ(t),V etc.). However, it could happen that the auxiliary data form a space with more than one component, e.g. when the ﬁbers are of dimension 2, then the choice of V has at least 2 connected components. Moreover, in general there is no guarantee for a Hamiltonian isotopy instead of a symplectic isotopy.

Remark 5.14. The tilted transport construction as described here can be easily generalized in many ways. A most interesting generalization is that one could admit V with singularities thus change the topology of Zt when t evolves. We will explore further applications of such constructions in upcoming work. 36 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

5.2.3. Construction of (-4)-spheres. Let us now specialize the tilted trans- P9 port construction to the case of −4-spheres in the class −a(−3H+ i=1 Ei)− 2E10. We will continue to use the notation in the previous section. 2 Take the usual Lefschetz fibration by elliptic curves on E(1) = CP2#9CP , which can be endowed with a Kählerform ω compatible with the fibration structure. It suffices to restrict the fibration to a neighborhood of a singular fiber, yielding a fibration over 0 ∈ D = D2(2) ⊂ C, 1 being the unique critical value and the generic fiber an elliptic curve. Denote by p0 the unique critical point of this fibration. Take 0 ∈ D as the base point b0. From the usual construction of vanishing cycles, there is a circle C ⊂ π−1(0) which has the following property. Consider γ : [0, 1] → D, γ(t) = t, then:

lim φt(y) = p0 ⇐⇒ y ∈ C. t→1

Here φt is the parallel transport using the induced symplectic connection along γ(t). Let π−1(0) be identified with a symplectic T2 = S1 × S1, where the two S1 = R/Z factors are parametrized by s, r ∈ [0, 1], and C is identified with {r = 0}. Take a neighborhood of C as S1 × [−δ, δ] ⊂ T2 = π−1(0), δ ¿ 1. Assume without loss of generality also that the symplectic orientation is given by ∂s ∧ ∂r, i.e. ω(∂s, ∂r) > 0. We propagate this coordinate to Eγ\p0 δ by the symplectic connection. Let V = 2 · ∂r, we define a tilted transport 1 0 γ 0 of C0 = S × {−δ} from E0 to E1, denoted as Σ0. Now C0 = ∂(Σ0) ∩ E1 bounds two symplectic disks on E1 (which is a fish-tail), but only one of 0 them concatenates by the correct orientation with Σ0. Concretely, this disk 1 δ is precisely the image of the usual parallel transport of S × [0, 2 ] to E1. We 00 0 00 denote this symplectic disk on E1 as Σ0. A suitable smoothing of Σ0 ∪ Σ0 yields a symplectic disk Σ0 with boundary C0 on E0. This is a symplectic variant of the usual vanishing thimble construction. Now choose another embedded curveγ ¯(t) ⊂ D2 so that  γ¯(0) = 0, γ¯(1) = 1;  γ¯0(0) = −γ0(0);  γ ∩ γ¯ = {0, 1}.

Again Eγ¯ inherits a symplectic connection thus a trivialization in the complement of the singular fiber and we can trivialize this part of the pull- back fiber bundle using the symplectic connection and parametrize it by coordinate (s, r, t) ⊂ S1 × S1 × [0, 1) as before. We may adjust the fibration appropriately overγ ¯ so that the vanishing cycle alongγ ¯ is again C = S1 × {0} × {0}. Let Wa = −(a − 2δ)∂r. The tilted transport associated toγ ¯ and 0 Wa starting from C0 gives a symplectic annuli Σ1 with boundary C0 and γ¯ γ¯ C1 ⊂ E1. C1 again bounds two symplectic disks on the singular fiber, and 1 00 we can take the image of S × [0, δ] under the (usual) parallel transport Σ1. STABILITY AND EXISTENCE OF SURFACES 37

0 00 The union Σ1 ∪ Σ1 again forms a symplectic disk with boundary C0 after smoothing. Now the union Σ0 ∪ Σ1 then matches to form a smoothly immersed sym- 2 plectic S , with adjustments on V and Wa near C if necessary. This sym- 2 plectic S is denoted as Σ, and it has a unique double point at p0 and is embedded otherwise. It is not hard to see from our construction of Wa that [Σ] = −aK for K being the Poincare dual of the canonical class of (E(1), ω), which is homologous to a fiber class: by resolving the self-intersection at p0, one has an embedded surface in E(1) which is smoothly isotopic to a multiple cover of a generic fiber of the fibration. One may then perform a small symplectic blow-up at p0 which resolves the self-intersection and which yields an embedded symplectic sphere with class −aK − 2E10 for any a ≥ 1 in CP2#10CP2 for an appropriate symplectic form . This proves Lemma 5.11 and hence the proof of Theorem 1.8 is complete. 5.3. Spheres with self-intersection −1,−2 and −3. To begin, we note that, when b−(M) = 0, there are no spheres of negative intersection. When b−(M) = 1, the only rational manifolds are S2 × S2 and CP 2#CP 2. Due to the existence of an orientation reversing diffeomorphism, the negative square case can be reduced to the positive square case. The minimal genus of A ∈ H2(M, Z) in these cases has been determined in [43] and from this all symplectic spheres can be determined. So we generally assume in the following that b−(M) ≥ 2. 5.3.1. Spheres with Self-Intersection −1,−2. Spheres with square −1 are exceptional spheres. They are all D(M)-equivalent to either E1 or H −E1 − E2 from [26]. We now consider spheres with self-intersection −2. A classification of smooth −2-spheres can be found in [26]. For rational manifolds M, La- grangian −2-spheres have only recently been classified in [34]. Proposition 5.15 (Lemma 3.4, Lemma 3.6, [26]). Let M be a rational − 2 manifold. Assume that b (M) ≥ 2. Let A ∈ H2(M, Z) with A = −2. Assume that A is represented by a smoothly embedded sphere. Then up to the action of D(M), A is one of the following: − (1) If A is characteristic, then b (M) = 3 and A = H − E1 − E2 − E3. (2) If A is not characteristic, then A = E1 − E2. This proves one aspect of Speculation 1.10 for spheres with A · A = −2. The following completes Speculation 1.10 in the −2 case for rational manifolds. Proposition 5.16. Let (M, ω) be a symplectic rational manifold and A ∈ H2(M, Z) such that A · A = −2. Then A is represented by a ω-symplectic sphere for some symplectic form if and only if

(1) gω(A) = 0, (2) [ω] · A > 0 and 38 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

(3) A is represented by a smooth sphere Moreover, when b−(M) 6= 3, V can be chosen to be the blow-up of an exceptional sphere. Proof. Assume that A is represented by a smooth sphere. Then A is D(M) - equivalent to one of the classes in Proposition 5.15. There exists a symplectic form τ with Kτ = Kst such that the classes from Prop. 5.15 are represented by a τ-symplectic surface. This can be obtained through an appropriate blow-up from CP 2. The result now follows from Lemma 3.12. ¤ 5.3.2. Spheres with Self-Intersection −3. We proceed as in the −4-case.

2 2 Lemma 5.17. Let M = CP #kCP with k ≥ 1 and A ∈ H2(M, Z). As- sume that A · A = −3. Then, up to D(M)-equivalence, A is a reduced class, −H + 2E1 or H − E1 − .. − E4. 2 2 Proof. For k = 1, a − b1 = −3 allows only for ±H ± 2E1. For k = 2, again Lemma 1, [27], reduces the problem to classes with 2a ≤ b1 + b2. This produces no further classes beyond the one above. For k ≥ 3, as in Lemma 3.4, [26], it can be shown using reﬂections along −2-spheres H − Ei − Ej − Ek that either A is reduced or 3 b2 + b2 + b2 − 3 ≤ a2 ≤ (b2 + b2 + b2). 1 2 3 4 1 2 3 2 P 2 In addition to this inequality, a − bi = −3 and b1 ≥ b2 ≥ ... ≥ bk ≥ 0. The solutions to this system, written in short as (a, b1, b2, .., ), are: (0, 1, 1, 1, 0..), (1, 1, 1, 1, 1, 0, ..), (1, 2, 0, ..), (2, 2, 1, 1, 1, 0, ..), (3, 2, 2, 2, 0, ..). Under the D(M)-action, (1, 1, 1, 1, 1, 0, ..) is in a class of its own when k = 4. The other classes are all equivalent, and when k ≥ 5, (1, 1, 1, 1, 1, 0, ..) is included as well. ¤ Theorem 5.18. Let M = CP 2#kCP 2 and k ≥ 10. Then there exists no reduced class A ∈ H2(M, Z) with A · A = −3 and Kst · A ≤ 1. Proof. This result follows by repeating the proof of Theorem 5.5 in the τ = 1 case. This involves no new methods. ¤

For k ≥ 3, −H+2E1 is equivalent to E1−E2−E3. This class is represented by a symplectic sphere. 2 2 Consider a symplectic representative V of −H +2E1 in CP #3CP . Blow up one point of this representative to obtain a symplectic −4-sphere Z in 2 2 CP #4CP in the class −H + 2E1 − E4. By Cor 3.3 in [6] there exist 3 disjoint exceptional spheres from Z. In particular, two of these exceptional spheres are just E2 and E3, both of which are not aﬀected by blowing down STABILITY AND EXISTENCE OF SURFACES 39

E4. Thus after blowing down E4, we re-obtain V as a symplectic manifold, but also obtain two pairwise disjoint exceptional spheres which are furthermore disjoint from V . Hence blowing down E2 and E3 leaves V unchanged, thus providing for a symplectic −3-sphere in CP 2#kCP 2 for k = 1, 2. Al- ternatively, the class −H + 2E1 can be related to a Hirzebruch surface; it can be viewed as a section in the non-trivial S2 bundle over S2. This is symplectic for an appropriate choice of symplectic form. We have the analogues of the results in the −2 and −4 case. Theorem 5.19. Let (M, ω) be a rational symplectic manifold and A a homology class with A2 = −3. Then A is represented by a ω-symplectic sphere if and only if

(1) gω(A) = 0, (2) [ω] · A > 0 and (3) A is represented by a smooth sphere Moreover, when b−(M) 6= 4, V can be chosen to be the blow-up of an exceptional sphere. 5.4. Discussions. We conclude this section on spheres in rational surfaces by indicating some possible directions extending further our results.

5.4.1. Spheres with large negative square. The tilted transport is used to prove that the classes of type (3) in Theorem 5.8 are representable by symplectic spheres. While the focus of the previous section was to construct (−4)-spheres, the results also prove the existence of highly singular curves in CP 2, as we will explain. Using Lemma 2.8, there exist exceptional spheres in the classes Ei which intersect the (−4)-sphere Va of type (3) locally positively and transversally and which can be blown down to produce a point of a-fold intersection. Doing this for all ten exceptional spheres E1, .., E10 produces a curve C ⊂ CP 2 in the class 3aH with one nodal point and 9 points of a-fold self- intersection. Applying Prop. 3.3, [33] to one of the a-fold intersections, one can succes- sively perturb away intersection components to make an a-fold intersection into a (a−1)-fold intersection and (a−1) double points. Repeating this will produce singular curves with diﬀering combinations of intersections. Blow- ing up at the intersection points will produce an embedded sphere in some rational manifold. This result is summarized in the following lemma. To each a-fold singular point, associate the value ki (1 ≤ i ≤ 9), 0 ≤ ki ≤ a−2 and a > 2, describing the number of curves which have been perturbed out of the singular point. Let k 1 N = k (a − i − ) i i 2 2 P9 and m = 10 + i=1 Ni. 40 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

Lemma 5.20. In CP 2#mCP 2 the class P X9 XNi A = 3aH − (a − ki)Ei − 2E10 − 2 Ei i=1 i=1 is represented by an embedded connected symplectic sphere.

Then X9 2 A · A = [ki − 2ki(a − 1)] − 4. i=1 This allows us to note the following interesting examples. (1) Consider the two classes

X6 X19 A1 = 12H − 4 Ei − 3(E7 + E8 + E9) − 2E10 − 2 Ei − E20 i=1 i=11

corresponding to a = 4, k1 = k2 = k3 = 1, ki = 0 otherwise and then one point blown up and

X7 X20 A2 = 12H − 4 −2(E8 + E9) − 2E10 − 2 Ei i=1 i=11

corresponding to a = 4, k1 = k2 = 2, and ki = 0 otherwise. Both classes can be represented by embedded symplectic spheres of self- intersection −20 in CP 2#20CP 2. Note that both classes are reduced, by the uniqueness of reduced form [26], they are not D(M)- equivalent. (2) It is very simple to construct spheres with self-intersection −l in CP 2#mCP 2 for some l > m. For example, the curve

X28 A3 = 9H − 2 Ei i=1

corresponding to a = 3 and ki = 1 has A3 · A3 = −31 and lies in CP 2#28CP 2 or

X5 X22 A4 = 12H − 4 Ei − 3(E6 + E7 + E8 + E9) − 2E10 − 2 Ei i=1 i=11

corresponding to a = 4, k1 = .. = k4 = 1 and ki = 0 otherwise has 2 2 A4 ·A4 = −24 and lies in CP #22CP . Compare this with the lower bound for spheres in irrational ruled manifolds obtained in Lemma 6.1. STABILITY AND EXISTENCE OF SURFACES 41

5.4.2. A local variant of tilted transport and symplectic circle sum. We explain next how to use a rather simple case of tilted transport to partly recover the circle sum construction in symplectic geometry. Note the corresponding counterpart is well-known in the smooth category. The setting under consideration is a pair of disjoint symplectic surfaces 4 ∼ S0,S1 ⊂ (M , ω). Suppose one has an open set U ⊂ M so that U = S1 × [−1, 1] × D2(2), a trivial bundle over D2 with annulus fibers, while 1 Si ∩ U = S × [−1, 1] × {i}. We claim that there is an embedded symplectic surface S which is the circle sum of S0 and S1. The question is local so we concentrate on the trivial bundle U. Remove 1 1 1 the part S ×[− 2 , 2 ] from the fibers U0 and U1. Consider two embedded arcs γ(t) andγ ¯(t), which only intersect at γ(0) =γ ¯(0) = 0 and γ(1) =γ ¯(1) = 1. By choosing W appropriately on Uγ, one easily constructs a tilted transport 1 1 1 1 which concatenates S × [−1, − 2 ] × {0} with S × [ 2 , 1] × {1}. Similarly one 1 1 1 1 concatenates S × [−1, − 2 ] × {1} with S × [ 2 , 1] × {0} by choosing another tilted transport onγ ¯. This realizes the circle sum as claimed. As immediate consequence of the construction, by taking a finite number of nearby copies of generic fibers in an arbitrary Lefschetz fibration of dimension 4, one realizes n[F ] as an embedded symplectic surface by performing symplectic circle sums on two consecutive copies. This construction clearly generalizes to higher dimensions in appropriately formulated cases, which is left to interested readers. This particular case applied to situations in Section 5.2 yields an alternative proof for Lemma 5.11.

6. Spheres in irrational ruled manifolds Let M be an irrational ruled symplectic manifold. Then the minimal 2 model of M is an S -bundle over a surface Σh with h ≥ 1. Recall that, when M is minimal, there can be no negative symplectic spheres from adjunction (see also the proof of Lemma 6.1), so there is nothing to investigate. In the non-minimal case, the blow-up of the trivial bundle and the blow-up of the non-trivial bundle are diffeomorphic, we fix a standard representation: Let 2 2 M = (Σh × S )#kCP . Denote by {S, F, E1, .., Ek} the standard basis of M, where S denotes the class of a surface of genus h and F is theP fiber class. Denote the standard canonical class Kst = −2S + (2h − 2)F + Ei.

6.1. Smooth spheres. Lemma 6.1. Let M be an irrational ruled manifold with b−(M) = k, and 2 A ∈ H2(M, Z) is a class with A = −l with l ≥ 1. Assume that A satisﬁes the following: (1) A is represented by a smoothly embedded sphere; (2) A · A = −l; (3) A · [ω] > 0 for some symplectic form ω with Kω = Kst and

(4) gKst (A) ≥ 0. 42 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

Then l ≤ k.

Moreover, up to permutations of Ei, X1−b Xl (6.1) A = bF + Ei − Ej, i=1 j=2−b for some b with b ≤ 1. P Proof. Write the class A = aS + bF + ciEi. Since the projection of any smooth sphere representing A to the base in an irrational ruled manifold is null homotopic, we must have a = 0 from condition (1) of Lemma 6.1. This 2 2 in particular shows that S × Σh and S ×˜ Σh admit no spheres of negative self-intersection for h ≥ 1. Conditions (2) and (4) of Lemma 6.1 imply that X X 2 ci = l and 2b + ci ≤ 2 − l which can be combined to give X (6.2) 2b + ci(ci + 1) ≤ 2. P Lemma 6.2. Let A = bF + ciEi and assume that A satisﬁes the last 3 conditions of Lemma 6.1. Then |ci| ≤ 1 for all i .

Proof. For n = 1 it is well known that A = Ei or F − Ei (see e.g [29], [32]). In fact, this classiﬁcation provides constraints on symplectic forms with canonical class KPst. Suppose ω is a symplectic form with Kω = Kst and [ω] = cS + dF + eiEi. Then since Ei and F − Ei are ω-exceptional classes, we have

(6.3) c > |ei|, ei < 0.

Case 1: Assume that b ≥ 0 and |ci| ≥ 2 for some i. Each term on the left of 6.2 is non-negative by assumption. In particular, ci(ci + 1) > 2 unless c1 = −2, cj = 0 for j ≥ 2 and b = 0. However, for symplectic forms ω satisfying (6.3), the class −2E1 is not symplectically representable, i.e. [ω] · (−2E1) < 0. Case 2: Assume that b < 0 and |ci| ≥ 2 for some i. Without loss of generality let c = 1. As [ω] · A > 0 we have X X b − eici > 0 ⇒ 0 > b > eici. for some c, ei satisfying (6.3). In particular, for some i it must hold that ci > 0. Then from |ei| < 1, X X X 0 < b − eici < b + ci + |ci|ei

ci>0 ci<0 STABILITY AND EXISTENCE OF SURFACES 43 and thus in particular X X 2 0 < b + ci and 0 < b + ci . ci>0 ci>0 This can be used to rewrite (6.2) as X X X 2 2 (6.4) 2 ≥ b + ci + b + ci + (ci + ci) . c >0 c >0 c <0 | {zi } | {zi } |i {z } >0 >0 ≥0 2 If there exists an i such that ci < −2, then ci + ci > 2. For ci = −2 we must have b = 0, this case has been considered previously. Thus we may assume that ci ≥ −1. In this case, the last term in (6.4) vanishes. As the ﬁrst two terms must be positive, they must both be equal to 1. Thus X X 2 (6.5) b = 1 − ci = 1 − ci ci>0 ci>0 which can be rewritten as X 2 (ci − ci) = 0. ci>0

This implies ci = 1 for those i such that ci > 0. We have thus shown that ci ∈ {−1, 0, 1} for all i. ¤ We now complete the proof of Lemma 6.1. The claim follows by reindexing {Ei} so that (ci) = (1, ··· , 1, −1 ··· , −1, 0 ··· ) and applying (6.5) to calculate b. ¤

The relation between b and ci given by 6.5 leads to the following three possibilities: (1) b > 0: Then 6.5 implies that b = 1 and Xl A = (F − E1) − Ej. j=2

(2) b = 0: Then there is a unique index with ci = 1. Thus Xl A = E1 − Ej. j=2 (3) b < 0: Rewrite A as follows: X1−b Xl X1−b X A = −|b|F + Ei − Ej = E1 − (F − Ei) − Ej. i=1 j=2−b i=2 44 JOSEF G. DORFMEISTER, TIAN-JUN LI, AND WEIWEI WU

Notice that in all cases, we can view A as having the class of the blow-up of an exceptional sphere.

6.2. Symplectic spheres. The stability results of Section 3 now allow us to conﬁrm the existence of a symplectic curve for a choice of symplectic form. In particular, we may ﬁx a choice of symplectic canonical class. This leads to:

Proposition 6.3. Given any symplectic form ω with Kω = Kst, and a class A ∈ H2(M, Z) of the form (6.1) up to permutations of Ei, A is represented by ω−symplectic sphere V if and only if A · [ω] > 0. Proof. To begin, we show that A as in (6.1) is D(M)-equivalent to either E1 − · · · − Ei or F − E1 − · · · − Ej. To see this, we consider different values of b: When b = 1, A is already in the desired form. If b = 0, then A = E1 − E2 − · · · − El. If A is characteristic then we are done. Otherwise, use the reflection along F − E1 − Ep to get to − F − Ep − E2 − · · · − El. Here p = b (M) − 1. Notice that p > l in this case. If b = −1, then A = −F +E1 +E2 −E3 ··· . Now reflect along F −E1 −E2 to transform it to F − E1 − E2 − E3 ··· . When b = −2, A = −2F + E1 + E2 + E3 − E4 ··· . Then reflection along F − E1 − E2 transforms it into E3 − E1 − E2 − E4 ··· . For b < −2 use reflection along F − E1 − E2 and induction to achieve this transformation. Now assume that A · [ω] > 0. Transform the class A via a diffeomorphism φ to one of the forms above. Notice that all of the transformations above preserve the canonical class. Thus the new class φ∗A is associated to some symplectic form with standard canonical class. Note that both E1 −· · ·−Ei and F − E1 − · · · − Ej can be represented by symplectic spheres obtained by an appropriate sequence of blow-ups for such a symplectic form. Now apply φ−1 to get the desired result. ¤ Corollary 6.4. Let M be an irrational ruled symplectic manifold and A ∈ H2(M, Z) a homology class with A · A = −l. Then A is represented by a connected embedded symplectic sphere for some symplectic form if and only if A is D(M)−equivalent to one of the classes in Proposition 6.1. Proof. Assume that A is represented by a connected embedded ω-symplectic sphere for some symplectic form ω. The transitive action of D(M) provides a connected embedded ω0-symplectic sphere for some symplectic form ω0 0 with Kω0 = Kst in some class A D(M)-equivalent to A. 0 0 Notice that gω(A ) = 0 and thus the result follows from Prop. 6.1. The converse follows from Prop. 6.3 and an appropriate choice of symplectic form. ¤ STABILITY AND EXISTENCE OF SURFACES 45

Theorem 6.5. Let (M, ω) be an irrational ruled symplectic manifold and A ∈ H2(M, Z). Then A is represented by an ω−symplectic sphere if and only if (1) A is represented by a smooth sphere, (2) gω(A) = 0 and (3) A pairs positively with ω. Proof. Clearly if A is represented by an ω−symplectic sphere the result follows. P Now consider the other direction. A = bF + ciEi as in the proof of Lemma 6.1, by considering projection to the base, thus we may assume that A · A < 0. Our assumptions imply that there exists a diﬀeomorphism 0 0 taking A and ω to a class A and symplectic form ω with Kω0 = Kst such that the above conditions continue to hold. Lemma 6.1 and Proposition 6.3 then imply that A0 is represented by a ω0-symplectic sphere. Now use the diﬀeomorphism to get a ω-symplectic sphere in the class A (see Cor. 3.8). ¤ This completes the proof of Theorem 1.9.

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